Skyrim for 360 freeze issues.

I acquired Skyrim through the 360 marketplace not too long ago. Only up until recently has my game, no matter location, save, and circumstance freezes and locks up the 360, until I get up and press the button on the console to turn it off. I've deleted multiple saves, deleted the cache, even deleted the game and re-downloaded entirely. But the problem persists!! Is this a sign for the imminent death of my 360? I have an older generation model but with the internal memory (latest one I believe), and it does not give me any other issues with any other game or application. If it's not my 360, is it a situation that will be handled? Do any of you suffer from the same problem? If any one knows a fix, please help. Many thanks.