The Game is linked to the device, you need to have a clean device (restore it to factory if you have a game already used on it).
once you have a clean device use the new account to start the new game. once the new game is on the new account you can redo everything again with your old account to get it back to normal and then just swap accounts with game center or play games.
I recently tried to link to an apple account on a friends device and it linked his game instead. we completely swapped accounts on the device, but didnt restore it to factory. still kept the last game on the device.
now how do you reset/delete just an apple game center account??
sigh.
P.S.
In android, the original account game will load first, no matter which game account you are logged into. you need to disconnect and reconnect within the game and it will pop up googles play games with a list of accounts. choose the account you want to play and it will say we have found so&so game in ???? Age.
say ok if its the game you want to play.
In Apple, the original account will load(show) but a pop up will say we have found so&so game... you need to change game center accounts in settings before entering the game to swap accounts.