The fat X-Box 360's overheat and die. it's going to happen eventually, to most people, if not everyone.
Not all fats are created equal. The Jasper models reduce both the CPU and GPU to a 65nm fab process (from 90), use a lot less power and generate a lot less heat. These are much more durable than the Xenons and Zephyrs of old.
OP, try a new character under an offline profile. If the tech issues go away, there is something wrong with the save files in your original game. If they persist, there is something wrong with the game install, the cached data or your hardware. If other technically-taxing games continue to work well, you can eliminate hardware as the cause. Make sure you try other 3D games that stream a lot of data from the HDD on the fly, though.