1992 Nissan Sentra Electrical issue

I have a 92 sentra and it has a wierd electrical problem. Its been draining the battery. The battery is fine if I let it sit for a couple hours but over night it dies. When the car is running and the battery is fully charged it makes this wierd clicking noise and the battery light and brake light start flashing of and on and eventually stay on. Also everything from the fan to the head lights and interior lights goes into turbo mode. I think the alternator is over charging and the internal regulator isn't switching off. And somehow the backfeed from that connection is draining the battery. I've had the alternator tested and it was ok. Should I just try a new alternator anyways?