For some reason, every time I try to play Fez without being signed into Xbox Live, it only let's me play the trial version, even though the only version in my game library is the full version. I'm trying to get all of the anti-cubes in the clocktower room by advancing the time on my Xbox (for which you must me offline). Now you see my dilemma. I could wait 12 more hours for the cube to appear but I'd rather not have to wake up at 5:00 to play Xbox. Although Fez is worth it. Help?
If the game was not originally downloaded on this system it will transform the license to a trial game. Although I've been having similar issues since last december with my games turning into false licenses. This is working knowledge from August 2012, when I purchased a license for a particular DLC then I signed off and my Girlfriend had the same license, same permissions and full execution of the DLC/Game. If you haven't done so, or if you don't remember where you purchased a particular item all Xbox users can use a license transfer which updates the licenses of all purchased items to your xbox or you could browse your purchase history to redownload Fez which would re-enable the license to allow you to play offline.
NOTE: This is based on old knowledge and I haven't had to try this in a while but should still be valid.
When you purchase a game on Xbox Live, two registration licenses are generated: one for the console that the content was originally purchased and downloaded on (as per "Cole Busby aka cheat117cole108"'s answer) and one for the Xbox Live account that the content was purchased with. You can download and use the content on the system with the "console" license using any account you wish: however, if you download/use the content on a separate console, you will need to be logged into the Xbox Live account that has the "content license" associated with it (which seems to be your current situation).
If you think that you SHOULD have the console license with the system you are currently using, deleting and redownloading the content may restore the license but be forewarned that depending on the game, this may also delete local saves: the process is detailed in solution 2. here: http://support.xbox.com/en-US/xbox-live/marketplace-and-purchasing/cant-play-downloaded-content
It is also possible to transfer "console licenses" to another system once every 4 months, as detailed in the "Can I transfer a content license from one Xbox 360 console to another?" section on this page: http://support.xbox.com/en-US/xbox-live/marketplace-and-purchasing/download-content#fc3c1f36d25f4dc2968fc8107dce18af
Seems this is a variant of an issue I had a few months ago. For completeness, the question that I asked is here: Xbox Gamertag Profile on Friend's Console - Can I download my content?