The game save is not related to your Premium file. Your Premium status is an account entitlement so your account is always premium as you can see on Battlelog, but in order for you to have access to download your maps for free, have the premium background, camos, and ACB-90 when you play the game you must have the Premium file present on your local Xbox 360 HDD. The Premium file is not linked to any other files in BF3 such as local game saves only to your Xbox Live account and Origin account.
Using a new Xbox Live account will not give you access to Premium since the entitlement is tied to your Original Xbox Live account that purchased Premium throughand it is through that old account that you must re-download your Premium file. If you deleted your old account from the Xbox account login screen that appears when your first start your Xbox scroll all the way to the right and choos "download profile" and then when it is downloaded you can open your purchase history and download your Premium file again.
Just to clarify the game save issue for you: Game saves on your Xbox 360 is for the single player campaign only. Your COOP and Multiplayer stats, unlocks, and entitlements like Premium are all stored off site on DICE's servers and not on your local Xbox hard drive so deleting a game save will not impact your Premium status..