Sounds like the alt could have a dead regulator.
To track down a current drain, remove one of the battery leads (-ve is safer) and put an ammeter between it and the battery terminal. Then disconnect things (or pull fuses) until the current draw drops to "not much". I'd start by disconnecting the alternator charge wire to see what happens. You should have less than 1 amp draw, give or take dependent on alarms, interior lights, stereo memory, etc.