Well, it worked! (Imagine that!)
My door had just two hinges, one at the top and one at the bottom. I took the bottom hinge off, cut six cardboard shims using cardboard from the back of pads of paper, and put three each on the door side and the wall side.
Aside from some technical difficulty trying to hold the shims and hinge in place while screwing it back in, once I got it in, the door catches fine. On the first try!!
Now I can keep my cat out of my bedroom, which will stop him from peeing in there! Now on to my NEXT problems - trying to get cat pee out of the carpet, followed by how to stop my cat from peeing on my carpet. I suspect these will be more difficult than shimming the door!
Tom, thanks so much for your help!