You own them, or more accurately, they own you. Don't worry so much about papers as taking care of them. You can contact the shelter and vet and ask if anything is needed to transfer ownership records but it is not likely this is a big deal since these are simple companions and not championship/registered felines.
I agree that you own them but I suggest you have them micro-chipped and registered (if your town requires it) in your name. You should also make sure that any "new" vet records are in your name.
If we do not have a signed fee agreement I am not your attorney and this is not legal advice.