A loose or weak connection can and often will over heat.
Even a brand new power cord and a brand new male connection on your
trailer can do this. MY power cord plugs onto the trailer very easy.
It's not melted anything--yet but if it did, I would not be surprised.
Sometimes when they mold those cord ends they can get just a tiny bit
of plastic in between the contacts in the female end and it can cause
a weak connection and result in over heating.
Don't freak out on not tripping your breakers until/unless you use an
actual clamp on ammeter and KNOW for sure what current is passing
through a particular circuit. Unless you use a real meter you are guessing.
It sucks that you need a new cord and socket! Remember the old boaters
mantra- A boat is just a hole in the water you pour money into.
RVs can be considered land yachts!!
Fix it, move on,