Gamerscore is tied to the Xbox 360 profile (local console profile), not the Xbox Live account itself. You cannot move achievements from one Xbox Live account to another.
If you still have the original Xbox 360 profile on the console, sign in with it. It will sync its Gamerscore to the Xbox Live servers under that account. Then sign in with your new account—each keeps its own score.
In extreme cases (account hacked, legal name change), Xbox Support might assist, but they generally do not transfer achievements between accounts. Don't count on it.
This applies to Xbox 360 only. Xbox One / Series X|S use cloud-based profiles—your Gamerscore follows your Microsoft Account automatically across consoles.