Well I think I found the problem! It turns out one of the wires to the alarm key switch was exposed and almost broken through from contact with the locking mechanism inside the door. I repaired the wire and put everything back together and the alarm is not chirping at me anymore!!! Here's a pic of the damage:
Fingers crossed when I get everything back together that I won't have any more issues. If so, I'll just leave the alarm unit connected as I found it.
Thanks again for the help!