GameCenter not Working?

Long story short, every few weeks, GameCenter stops working.

No games will load GameCenter, and if I press GameCenter, it just goes onto a white screen. If I go into GameCenter in my settings, it will have my username, and a constantly loading swirl under it.

I've been forced to reset my phone (not settings, as that doesn't work, and honestly, it annoys me how many settings it changes), multiple times now to fix it, and I'm just done doing that.

Does anyone know if there's a way to fix this without resetting my phone? (And even then, using the backup that I make AFTER the issue starts still lets gamecenter work, so I'm not even sure how it's broken.

Any help is appreciated. ._. This has only happened since iOS7, btw, and it usually occurs around the time a new OS update is available.