Try googling "pet adoption contract". The two I checked had conditions like not keeping the dog on a chain, promising to keep vaccinations and health checks up to date, etc. Doesn't seem out of the norm or impractical. I think I'd look your rescue up on the web and see if they have an agreement posted that you can review.
Based on what you feel might be strange behavior by your contact, I'd definitely review anything you sign but if you choose not to sign their agreement I doubt anyone's going to come after you.