here is a summary from many people with long experience:
Huntingdog: The most common cause is a bad connection at the plug
Camper-don: Check and make sure the receptacle at the campsite pedestal is in good condition
ScottG: The receptacle was worn and damaged your new cord/plug.
Time2roll: Typical of a cheap end crimped on. Get a premium 30 amp plug replacement.
Lynmore: With a 30 amp breaker protecting the outlet and a 30 amp breaker in the trailer, it would be impossible to overload the plug.
There was a high resistance connection that caused the heat, plain and simple.
Spridle: Bad luck at two separate camp grounds with bad receptacles at the post.
Ehoefler: Most likely you are doing it at home when you plug into the oulet with the adapters.
All of these statements have merit. It seems the culprit is the campground receptacle, with a slight outside chance you have a problem at home with a adapter or loose receptacle. My guess is bad receptacles at the campgrounds.
You do need to replace your male 30 amp cord cap. It is ruined. 12 dollars on Amazon. You dont need to replace the entire cord, just the male cord cap. Copy and paste into the Amazon search engine.
Camco Heavy Duty 30 AMP Mini PowerGrip Handle Male Plug Replacement