Ignore the volt gauge in the cluster for now...what voltage are you getting at the battery with engine on?

As far as other causes...I'm starting to think the CPU (maybe the alt warning relay in it) is the cause. It may be even some soldering joints or a bad connection, since you presumably would have had to touch those plugs in a swap? If you have another one handy or accessible, try swapping that one in. Anyone nearby in Texas?

Could be also an iffy warning light cluster. BTW, the only constant beep-buzzer I can think of is the seat belt buzzer that works that usually works exactly opposite to what you have, when you turn the ignition on. Any diff. when the seat belt is fastened? Just a thought.