Battery drains are difficult to track, yes, the alternator is dead, but it could need a simple Voltage Regulator for around $40
GET a multimeter and take some battery readings with the car off, key on, car running, and report back with what you find.

__________________
John HAUL AWAY, OR CRUSHED CARS!!! HELP ME keep the cars out of the crusher! A/C Thread
"as I ride with my a/c on... I have fond memories of sweaty oily saturdays and spewing R12 into the air. THANKS for all you do!

My drivers:
1987 190D 2.5Turbo
1987 190D 2.5Turbo
1987 190D 2.5-5SPEED!!!

1987 300TD
1987 300TD
1994GMC 2500 6.5Turbo truck... I had to put the ladder somewhere!