Got a chance to look at it a bit better today, and found the problem. It appears a rodent built a nest in the empty space in the tail behind the taillight and spent some time chewing on the wires. I pulled the tail light cover and the reflective piece in there, and, when I pulled the sockets out, I saw stuff packed behind the socket and the wires on the right-hand socket were chewed up.
At this point, it looks like I have to pull some more stuff off to get access in there, but at least I know what the problem is, or, more specifically, where it is. I'm sure there's just stuff touching in there that shouldn't be.
Thanks, again, for the help. I was scratching my head as electrical stuff is not one of my strengths and I appreciate the advice I got.