When it gets hot that indicates a short. Does it get hot with engine off and key in "run"? The alternator output goes to the PDC, which is all fused, the battery and starter, the latter 2 are supposed to have fusible links, but may have been replaced with standard cable. Sooooooooooo. It is possible the short is internal to the alternator. I find it interesting the engine runs when the alternator gets hot and is not charging. That means it is running on the battery, how long has it run in this condition before you stopped and felt the alternator?
I think we're going to need a bit more info to steer you in the right direction.