Update:
This condition got worse and worse until the speedometer just stopped working. Naturally the weather turned foul and it's been under 30 degrees for the past few weeks.
I just got a chance to climb under the car and look around. Just as Tim said, the wires leading to the diff became very brittle. When I touched the wire, it broke where the wire connects to the 90 degree connector. As a temporary fix until I get a new one, I was able to take the connector apart and get at the 90 degree connector. I then took a knife and whittled away at the plastic strain relief part until I got to fresh wiring. I soldered in a new wire and taped it all back up.