Thanks, all. I'd already thrown out the first wire, so I unfortunately don't have any pictures of how much melting there was (about 2-3 inches!), but I suspect that there may have been abrasion which led to further melting. The border between the wire and the insulation was very smooth, so it was definitely melting.

Attached are pictures of the current wire, which is also starting to show through the insulation a bit. Hard to see, but the border is rougher, so perhaps there was abrasion that's leading to a short on the frame. However, this wire had much more slack than the first one, which leads me to believe that perhaps because I wasn't using the spades I was getting a short elsewhere. I found that the spades were quite delicate, and after they broke on a tour I simply stripped the end of the wire and pushed it through the holes on the connectors. When I get new wires I'll be sure to utilize the spades and see if the short is still happening.