This solution is only if you bought the game on one console, and are now trying to play the game on another console (whether due to RROD, upgrading to Elite, or other reason), and are logged into Live, but the game is still in trial mode:
1) Start the game as normal
2) You will see you are in trial mode.
3) Go to Unlock game option in main menu.
4) Choose to download the full game. Wait until it's done.
5) Important next step, or this won't work! Choose Continue option (NOT Play Now option), and then choose cancel (B Button).
6) If you did step 5 properly, you will return to the game, and after a second or two, the "Unlock Game" will disappear, and you can play as normal (i.e. full version).
Note: If you choose the Play Now in step 5, the game will reset, and all your hard work is for nothing.
Note #2: Yes, you have to do this EVERY time you completely quit the game and then start it up again.