First off, the alternator needs to be replaced. It doesn't matter how old it is, it failed. Any short that would have caused this would also drain the battery overnight. I would replace it, the fusible links between the relay and the alternator, and the battery cables (both battery and the relay to starter). The fusible link and battery cables are cheap, and should be replaced regularly anyway. I'm sorry if this isn't what you want to hear, but based on your description, there's no doubt.
Take care,
~Chris
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