Is it possible to change the email of the Game Center account without changing your account entirely?

The Game Center is just the Apple ID. Once you change the Apple ID via this site all you have to do is log out of your Game Center account in the settings and re log in with the new email.

you will see all your achievements are still there and the games will be where you left off. And your username will stay the same.

Maybe related to this?

Community Specialist post about de-linking Game Center from unknown Apple ID --> https://discussions.apple.com/thread/253243345