Replace the drywall but preserve the molding trim?
I wouldn't use a recep saw. Seem like you want to start with a prybar and depends on how its starting to look you can then use whats called an oscillating multi-tool. This will give you a much cleaner cut with more control. It's a great tool to own. 
My guess is those guys used 2-3 inch nails which is a pain in the butt to remove. Had to go through this with my 1970 house.