Can a bad alternator run a battery down?

Looking at some data for your vehicle it shows a 30 amp fuse between the alternator and the battery. I assume it is the main output lead of the alternator. The fuse number is 26 and it is the fifth fuse down in the row furthest from the relays in the same panel. From what you say I think you may have found the trouble and will need to replace the alternator if that is correct. Blocking diodes can go bad inside the alternator and cause this kind of trouble.