Game Center constantly asks for password, forgets login
After updating to iOS7 (and now iOS7.0.2), Game Center always pops up right after I open any game and asks me to log in. It prompts me for the password for my old Apple ID (which I haven't used in over a year and for which I cannot recover the password). When I type in my new Apple ID and password, they are rejected with a "Cannot log in to Game Server" message.
When this happens, I go to Settings > Game Center and put in my current Apple ID and password AGAIN. This temporarily solves the problem, but inevitably Game Center will pop up and ask me for my OLD Apple ID and password.
I am guessing that there is a preference file somewhere which is remembering and asking for my old Apple ID and password despite my best attempts to teach it the new one. How do I correct this problem?
iPhone 4S, iOS 7.0.2