Alternator question, battery is discharging quickly.

[FC Zach]

For a while until this Monday, I have been dealing with some mystery drain to my battery. I found it by accident while detailing the engine bay where I heard a faint electrical whine that seemed to come from the alternator which stopped after removing the battery lead.

Before finding this issue, I'd have to keep a trickle charger connected to the battery or else it would completely drain within a couple weeks. It was attached when I found this problem. I left the lead off the alternator and disconnected the charger and left alone for a few days.

After giving it a few days with the charger and alternator disconnected, I reattached the charger to see if it would attempt charging the battery but it didn't. It was fully charged so I can confidently say it was the alternator.

Details:
-'87 RX7
-3rd Gen Alternator
-The 2 pin connection on the alternator is vacant
-Battery is good, tested. Tried another in case.

My questions:
-Does the two pin connector need to be connected to the alternator?
-If it's not a bad alternator (going to test it soon), any recommendations on what to check next?