Well, Jeremy; I've read all the info here now, and have some thoughts as to the direction I might take, and a quick word on the quotes you've been given.
On the $20K -$25K figures you've been handed; based on the work to be performed that you describe, and assuming that overall this is supposed to cover the full spectrum, including your plumbing & any needed electrical re-work...I would only say that, honestly, those figures don't sound off the scales or surprising, but within the scope of reason for professionally completing & warranting the work you describe. To physically detach & raise a 2 story home, move it off it's foundation (2' or 20'= basically the same difference), re-work/restore/rebuild & extend (in height) the entire footprint foundation, and likely one internal central bearing wall or columns & finally re-setting the home and re-connecting services is a large undertaking, with a lot of risks and variables. I know $20K still can be a "chest-grabber," but there are many aspects of this kind of work that will often run into "the big bucks."
Now to the task at hand;
Note: The folowing all assumes that the home is not a brick exterior, but sided in some other fashion. Since contractors were discussing moving the home, I'm sure this is the case.
First, to address some concerns on a couple things you mentioned doing;
-unless you were planning to use them horizontally stacked (alternating), log-cabin style, in a 4-sided box fashion to create large support points, not as jack posts or horizontal bracing above jacks, railroad ties are out. Don't use them, please; they will vary GREATLY, not just in straightness, but more importantly in their individual soundness, and can fail without warning. Preferably, you would use stabilized jacks under steel I-beams set horizontally the full width of the home perpendicular to, and directly below the joists; they can be bought relatively inexpensively, used from steel salvage co's, or new , pre-cut to lengths you specify. When you're done with them, you sell them back to the steel salvage. (The I-beams are directly under the joists, with the jacks lifting the I-beams.)
-the footing size you mention, 18" X 6" is not large enough to meet the needs of a 2-story structure. Minimum footing size in my area for a single story home is 20" wide X 10" thick; I've no idea what the code requirements are in your area, but I'd imagine you'll need 24" wide X 12" thick. The exception to this is that I think you were speaking of a new scondary footing, not replacement of the existing, so the width may be okay, but I'd still go for 10" thick as a minimum.
As to what is the best way to handle the overall project, personally I see 3 reasonable options;
1. Raise the home in place; straight up, with a jack system & the I-beams from below. Remove both the inner and outer, original foundation walls, and rebuild using either block, or have new walls "slip-formed." In this case, that is essentially new concrete walls poured against the dirt on the outside, with traditional forms only being used to the inside.
2. Raise the home as above, remove the interior, secondary walls and only the top few feet of the outer walls. Pour footing extensions to the inside, form the interior 8" inside of the original foundation walls and form the exterior up from the outside of the remaining original wall, and pour new walls right against the remaining original walls, thickening above them to replace the removed section at the top.
This last option is different entirely;
-it reinforces the existing, original foundation walls.
-is far less costly, requiring no large equipment logistics or clearances outside the home.
-could almost entirely be done privately (with your general background in construction).
This option first assumes that the primary problem, which is also the most common, is not so much the vertical, compression strength of the foundation walls & their ability to support the home, but that exterior ground movement and pressures have deformed and displaced the walls towards the interior space, yet not so far as to make them beyond structural value & use if reinforced. It also assumes that the original walls, although displaced, are still generally in tact and of reasonably sound make-up (major portions of the wall's individual bricks haven't broken down to crumbs, and mortar joints, although needing repair, have not been completely lost.)
This method is extremely effective and permanent on poured concrete walls, however, the placement of reinforcement on poured walls varies with crack locations; in this instance being brick, the placement will need to be more regular, with exceptions to any specific points that might show greater overall movement, in which case reinforcement of the specific area should be installed in addition to regular placement. In other words, if there's a specific point on a wall that is failing, you would reinforce directly on that point in addition to the regularly spaced reinforcement.
Now, while this method will stabilize the walls, it does not provide for any permanent change in the height (elevation) of the structure, nor would it (by itself) prevent or stop groundwater from entering the basement. This method, by itself, will simply halt the failure of the foundation walls, and restore their ability to do what they were meant to do; provide a solid base for your home. Any desired or needed preventions of water intrusion and any completion of interior finishes can be done separately at any time should you wish to do so.
Option 3. Begin to raise the home in the same manner described earlier, but in this case only enough to relieve most of the structural weight from the original foundation walls. Once supported, remove the secondary walls and inspect the overall condition of the original walls, and note any specific problem areas.
With any specific points being extra, on each original wall, at 8' intervals, but beginning no more than 4' from outside corners, you will be placing vertical 4"x4" I-beams that will nearly match (1/2" short is okay) the height of the space from the bottom of the joists to the basement floor, each having a 1/4" thick 4" X 6"steel "shoe" plate welded to the bottom of each beam, flush to one flange, with the added 2" of the shoe pre-drilled with a 5/8" hole, placing this side towards the inside of the basement, with the back side of each beam in contact with the innermost point of the wall that still allows the beam to be vertically level. With the beam held in position, mark the hole on the floor, set the beam aside, and drill the floor centered on this mark with a 3/4" concrete bit, 3-1/2" to 4" deep. Pour the drilled hole full of anchoring cement, re-align the beam, and then drive a 5/8" X 3" high strength bolt (with a smooth upper shaft, not threaded all the way to the head) through the shoe into the hole until seated. At the top of each beam that's along the walls where the floor joists are perpendicular to the wall, pre-drill, then lag bolt sections of 4" angle iron long enough to span at least 3 joists when centered on each beam, to the joists in front of each beam, with the downward flange tight against the inner face of the I-beams. On the walls where the joists are parallel to the wall, install full height 2X cross-blocking between a minimum of the outermost 6 joists, in sets of 3 on 16" centers, with the middle row centered on each I-beam, then bolt angle iron sections to these blocks in the same fashion as the other walls. Once all beams are secured top and bottom, make up a stiff mix of full flex mortar, and completely pack & fill any space between the back of each beam and the walls, top to bottom. Remove your jacks; the floor joist structure of the home now is on duty, if needed, to act as a "spreader" in horizontal compression, working in tandem with the I-beams to prevent your outer walls from collapsing inward any further. From here, you can also do waterproofing treatments to the walls, etc, as might be needed, or leave them for a future time.
There are a couple last, loose ends I'd like to cover here; the first being reinforcement of any added concrete work you might do.
Above, I've not made reference ('cause I planned to now, to simplify things) to rebar reinforcement in any & all of the described processes above which involve newly poured concrete. If you were able to pour new walls on top of your existing footing, you'd want to first drill your footings and install vertical rebars with anchoring cement, similar to as described for the vertical beam floor bolts, or per the requirements of your local building codes, then adding the horizontals also needed. If you decided to extend the footings for an added new wall inside the original, I'd drill into the existing footings and install horizontal rebar "pins," which would connect the old footing with any extensions. In short, any new poured concrete work that you will do here, should also include the required and/or appropriate rebar/mesh reinforcement.
I also wanted to briefly "wave the caution flag" to you in terms of taking on any actual raise or lift on the house. Even though you could feasibly do so, and everything MIGHT go just fine, you could also find yourself in a position where something, maybe not completely disastrous but costly, goes wrong during this process; in accepting responsibility for the work, you've also assumed the liability, and should something go awry, you could find that your homeowner's insurance will not only refuse to cover any damage, but could also decide this was cause to terminate your homeowner's insurance, leaving you out in the cold; just a warning here- insurance co's are really getting tight these days, and will use any rule, slant, or stance they can use- moral, fair, or otherwise, to keep every dime they get without giving any back, and should be also considered by you beforehand, just in case. If you're doing this work personally, especially without a permit, and something goes wrong, you're insurance can & will find a way to deny coverage, and drop you to boot. Cover all your bases.
Lastly, for the moment, I'd suggest that you do some further preliminaries before taking this on; One, I'd open a few holes in that secondary wall to get a glimpse of what general condition the original wall is actually in. Next, I'd decide what angle of attack I would follow, then have a structural engineer make a visit to the home, so you could show he/she (yes, one of the sharpest S.E.'s I've ever worked with is a woman)the existing structure, and discuss your intended restorations. Even if all you get from this meeting is a green light to your plans, it's worth it. 12 years ago, in K.C. MO, I raised the entire interior (foundation line was good, but the whole home sagged inside of the foundation) of a circa 1903, 1300 sq. ft. (footprint; ground floor footage) 2-story baloon-framed home a full 6" to restore level floors. Previous engineers had been called in, and each had told the owners to brace the home where it was & that restoring level was not structurally feasible. I looked the entire home over very closely, not cosmetically but skeletally, and after a few weeks thought decided that if it were done right, this was not impossible. I then put my thoughts to paper, expanding on them in detail and formed a plan to raise it and restore level. At this point, I knew I'd done my homework well, and had no real reservations, but I still called an engineer to come take a look and discuss my plans. Believe me, when you're wife & family is standing in your yard, scared to be in the house, talking to you on a cordless phone in the basement, underneath tons of material which you are now putting in motion on a bunch of seemingly tiny hydraulic jacks, and everything above you is creaking, popping, and making not uncommon sudden movements, where framing which has been slightly separated for years quickly re-seats with a loud bang, you will feel more secure knowing that someone who's licensed & has been professionally trained in this area has given you the "thumbs up." Oh, you'll still have that "searching for the nearest exit" feeling, but it's under more control because you've had your plans and methods o.k.'d by a pro in that field.
Even if you keep the project to a minimum, this is no small undertaking, so consider all your options very carefully first, Jeremy; only then would I jump in to a plan of action. If you'll actually be raising the house & changing the finished height of the home, look into having a licensed & insured pro take on this aspect of the job.
Jeremy, all of these options are viable, tried-&-proven remedies for your situation. I've tried to break down into fairly specific detail any parts of these methods which I thought necessary or of added assistance; don't hesitate to ask further if I've not been clear on something, or you have more questions; this original explanation was the "biggie"; anything further needed will likely be a piece of cake by comparison.
Again, I hope I've covered everything in an understandable way; if not and you'd like, let me know & I can get a CAD drafted viewpoint sent to you for reference. Hope you'll find some of this info helpful, inspirational and/or worth having waited for; Tom was right in expecting my thoughts on this one, but likely nobody might have guessed I'd submit a small novel, LOL.