Mike, I would try one of the extraction type of upholstery cleaners which apply the foaming cleaning solution and immediately suck it away. I've used these on vehicle interiors, and the seats are left very clean and almost dry enough to sit on. Then follow this up by plain water and the extraction to leave the cleaned surfaces damp, but not drenched. Then a little Texas wind and sunshine to finish the job. Hopefully, the linings aren't attached with a water soluble adhesive.
I'm assuming that the amount of damage is probably not not much more than your insurance deductible, or you'd just be shopping for replacement gun cases.
