Is there any way to debond construction adhesive without damaging cork or foam?
The key part of your sentence is “without damaging”. A putty knife, wielded very carefully, might allow you to do so, but I kind of doubt it.
Latex caulk can be worked off in this way, but not construction adhesive. Mineral spirits would also probably work, but they’d certainly damage foam and maybe cork as well.
Your best bet is probably to use the putty knife and salvage what you can.