I got this procedure from an online chat with Apple support:
Open the Game Center App. Login, then log out.
Force close the Game Center app so it's not in the background.
Open your game and close Game Center. Close the game app and then force close it from the background. Repeat this 3-4 times until it (game center) does not come up when you open your game.
Go to settings, and log out of the app store.
Turn off your phone.
Wait 3-4 minutes.
Turn on your phone.
Open your game and Game Center should remain off/disabled.
Did this procedure earlier today and no game center on my games so far. Hope it lasts.
edit:
I logged back into the app store with no ill effects. Game center remains disabled...so far.