Well, depending on what's under the shingles, you may be able to 'remove' it, then re-insulate, then re-sheath. YOu'd want to do this in very small 'sections' at a time because of the 'stresses' that the studs bear. The sheathing actually helps keep the house from collapsing like a 'house of cards'.
If you have the room on your lot, do as much demolition as you can BEFORE you 'order' your Hazardous Waste Dumpster. You pay by the week. By piling all the refuse in the yard, you save yourself some $$$. ANd when you finally order your dumpster, you can 'describe' the amount of refuse you have to the Hauler. (I suggest you ask that they add 25% capacity. You may want to empty out the garage or shed or basement too, or get rid of some cinder blocks or tree stumps or that old picnic table, if you know what I mean.) Also, IF you were to have that dumpster sit on your lot week after week, don't be surprised to wake up and find 'stuff' in it from your sneaky neighbors! It does happen. So by waiting until the 'last minute' to order your dumpster, you save more than just $$$. Oh, if there's extra room, THEN you can have your neighbors pay you a few $$$ to use up the space. BEFORE you call the Hauler to order your dumpster, roughly measure the sides of the house (and roof and whatever) because when you describe what you have to the Hauler, they do it in squares. In other words, if one side of your house is 30' x 20' = 600 sq. ft., that's 6 squares of shingles (and maybe sheathing in your case.) Be SURE, again, to tell him all the other 'stuff' you have to throw in.
Yes, definitely, you have some humidity in the house that's condensing on the cold toilet tank(s). Just as an aside, we have a 27 year old Colonial where the bathroom in the Master Bedroom has 2 outside walls. And the toilet is in the 'corner' of where the 2 outside walls meet. We have R-19 in the walls with vinyl over Gypsum Board. (I hated the Gypsum but I had no need to remove it.) Anyways, with that setup, I have no condensation on the tank. Wouldn't it be 'nice' to look forward to that when it's all said-and-done???