I can see that there are plenty of iOS users looking for a way to fix this maddening problem. After logging out of the current Game Center then Signing In with other one load another Clash of Clans village, the old village is still loaded, even after doing restart or reinstall the game…
Troubleshooting: Clash of Clans village gets stuck on iOS 7 and 8
If you are looking for a way to load other village or start playing a new fresh one without losing data, just follow these simple steps (thank cossythisisart for sharing!)
Connect your iOS device to PC/Mac and Backup it with iTunes.
After finishing Backup, open Settins ~> General ~> Reset ~> Erase All Content and Settings ~> Erase.
After resetting, choose Set up as New.
Sign In with the account you want to load another/new village.
Download Clash of Clans from App Store.
If you want to load an existing village, press OK to load it. If you want to start playing, finish the opening tutorials and name your new village.
Factory Reset your iOS device again, this time choose Restore from iTunes backup at starting.