If you go around and add up everything in your 30AMP trailer you will find it is more than 30AMPs (3600WATTS)...
I sometimes have problems with mine by turning on some the very high wattage items at the same time and tripping breakers...
We sometimes use the 20AMP outlets at the camp ground pedestals and run an extension cord into the trailer to run a few thing like portable heaters, hair dryers, etc to keep the load down abit from the 30AMP side... This is when we are really using the air conditioner alot...
If you have burn marks around the shore power connections you really should have them replaced. Nothing should get hot enough to do the damage you are referring too...
Roy Ken