If the overheating is occurring at the shore power receptacle, then there's a good chance the problem is in the receptacle. Low voltage could be the problem but I'd say there's a better chance the problem came about because of a bad connection(s) inside the box or a worn out receptacle. The male plug on the shore power cord definitely needs to be replaced if it got burnt/melted.
We just encountered this problem back in May. Plugging into the campground's receptacle, my male plug engaged a bit too easily. Had a feeling it was going to be trouble so I made sure to keep an eye on it. Sure enough, within about 30 minutes, I noticed my plug starting to get warm and the electrical box getting hot to the touch. I immediately turned off the breaker and disconnected. Got campground personnel to install a new receptacle and everything ran cool after that.
03 Itasca Sunova, Workhorse P32 with the 8.1 and 4L85-E