Actually, I'd probably be more confident doing this:
1. Clean the seam edges with MEK to get rid of the dirt, then rough it a little with a rasp
2. Take some Gorilla Tape
to the worst of the seams, and use a roller and a heat gun (carefully).
Go nuts with that tape- it'll hold pretty well if you have enough of it.
That might actually have a better chance of holding than trying to clean all the old adhesive from in between separated material & messing around with applying new glue. Seriously, either Stabond (PVC) or some kind of urethane glue is going to get very expensive. It's $30 to $40 for a pint of the stuff. Add in some patch material, and you're a 1/3 of the way into a set of decent cat tubes.
I'm not trying to be an ass - I just don't recommend sicking $200 into a lost cause.