I have had to remove old siding for various historic projects and reinstalled. For me to assist you I would need much more info and pictures to give you an idea what is the best approach. Doing this wrong could damage your old house. We can use this forum or you can contact me direct thru my website www.rmdesignconst.com Being so busy saving our history I do not have time to visit this site often. I have help many forum members thru the years with all kinds of questions
First of all, it is not the insulation that is "causing" the problem it is liquid water that is entering the wall from somewhere. So the goal is to find that source of water and eliminate it and block it from damaging the framing/insulation/plaster etc. You don't mention what type of siding it is so it is impossible to know if the siding can be saved. In terms of sheathing, wrap there is the Zip Wall system which is basically water resistant sheathing with the joints taped- eliminating the need for housewrap [which is mostly an air barrier rather than a water barrier- you need both.