Howdy–
Well, you're probably looking at the run cap. It'll have three connections on top, one labeled HERM, one fan, one common. It'll be something like 35-5 or 45-5 or something. You need a good run capacitor, to run your A/C. You can also add a start capacitor, especially for hard-starting compressors or to assist where voltages/amps are questionable. You only need the start cap for the compressor, not the fan.
Melted or damaged plugs should be repaired now. They only get worse with time. Poor connections cause resistance, resistance causes heat, heat causes fires. This is bad. You need to either repair this yourself or get an electrician to repair it. If the plug is bad, you could be getting improper voltages to your trailer & A/C unit. This could be the source of your whole A/C problem, and could also toast other expensive stuff that you like.
I wouldn't fix anything else right now except the male & female plugs. If doing it yourself, be sure to strip the wiring back to where no discoloration or damage is. And buy high quality replacement plugs.
Good luck, and before you do electrical repairs, and you are not sure, ask an electrician. Wiring must be done right, to be safe. And, when wiring, I never have a beer until the job is done.