Coby, I used to be a carpenter in N.O. and I shored many houses. You can do it yourself, particularly if it is a one story house.
You need to level the house before you attempt any interior repairs and painting because the interior structure will shift.
Make certain that you can access all of the areas that need shoring. Specific problem access areas include front and rear porches or stairs, particularly if they are poured concrete.
Find the highest part of the house. If you are happy with the foundation at that point, then you will raise the rest of the house to that level.
You will have to rent or buy a few or perhaps many jacks depending on the size of your house and the amount that you need to move it. Screw jacks are the safest. Hydraulic jacks are easier to use but risky. Be sure the jacks are rated to support the load.
You will also need to obtain railroad ties or minimum 6x6 beams (probably larger) to act as foundations for the jacks and to prop under the floor joists/beams/sills and above the jacks.
The jacks should be perpendicular to the ground. When you first start jacking, the beams under the jacks will be pushed into the spongy soil before the house begins to lift. Be certain that the jacks remain perpendicular to the ground during this stage.
Use a water level (the kind with plastic tubing)to level the different parts of the house.
If the brick piers are sound, then insert pieces of slate between the brick piers and the floor beams/sills. Slate won't rot or compress like wood.
Prop up safety supports in case the jacks slip.
Reset or preferably rebuild the leaning brick piers. Frost is not a major issue!
After you finish you may need to dump fill dirt under the house and divert all gutters and drainage away from under the house. This is the primary reason that the piers lean in New Orleans. (Or perhaps it is from one too many beers or Hurricanes.)
One time I was jacking up a three story house in N.O. from the basement. The floor was concrete and it cracked suddenly. It sounded like a bomb and the jack and post on top of it went flying. Luckily no one was killed. Be careful if you are jacking lots of weight.
Go look at other houses that are being shored. Tell your shoring contractors that you want to see their work.