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Checking for voltage between the center contact and the outer wall yields a measurement of 14.18v on both sides. However, the resistance between the outer wall socket and the body ground exceeds 1.552 M (Mega) ohms on the right and is 116.2 K (kilo) ohms on the left side. The right side bulbs fits decently snug but the left side does not. This was likely due to previous damage or damage I caused with the needle nose pliers when removing the old bulb assembly from the socket. -Grant
With the fact that you can read 14V between the light socket contacts, it appears the socket has power being supplied to it. That pretty much leaves the bulb connections with contacts in the socket. Check to see if the center contact of the socket hasn't been bent down when you removed the old bulb base.
Unfortunately, I haven't yet had time to look at this socket anymore this week. I will post an update sometime this coming week to see if messing with the center contact will fix the issue.
Update: I tried messing with the reverse sockets more but was not successful. In the end, I ordered two new ACDelco LS39's and spliced them in with the proper connectors. The reverse lights now illuminate perfectly. Thank you to all that responded and helped me diagnose this issue.
Did you find anything in the old housing? Crossed wires?
I did not unfortunately, my belief at this point is that the outer metal ring inside the assembly was too damaged to contact the bulb properly. Removing the original bulbs that had been left out to the elements took quite a bit of force and I don't question the fact they were damaged in the process.