I would recommend replacing the compressor capacitor.
Also get a Kill-A-Watt from Amazon.com they are about $18. You plug this into one of your RV receptacles (mine is in the kitchen where I can watch it while running the A/C unit.) If below about 107 volts, don't run the A/C there is a problem.
If the voltage is say 118 volts, and running just the fan brings it down below about 111 volts, then you can not safely run the compressor, as the voltage will drop even farther. The root cause is the 30 amp plug you are plugging into is very weak, overheating your plug end, and melting the wires. You probably only had 100 volts to the compressor when it was cutting off, due to a 10 - 15 volt drop across the plug to the camp post.
What I did was I was living in a campground with only 50 amp receptacles, so I changed my cord end to a 50 amp. I can still only use up to 30 amps due to the main breaker inside my RV, but the 50 amp plugs are in much better shape than the 30 amp plugs. (that get overloaded, and the brass contacts inside become weak).
So I would do two things. Buy a 50 amp plug for the end of your cord. Any hardware store will have them, it is the same as a 50 amp home stove receptacle plug in. RV parts store will have the same thing - more expensive probably.
Get a 50 to 30 amp adapter, for use in RV parks without a 50 amp service, and a 30 - 15 amp that you probably already use when in your home.
Then order the Kill-A-Watt. If you find that you typically camp in areas below 110 volts with the A/C on, you probably will also need a Hughes Autoformer, it will boost the incoming voltage by about 12 volts when needed (if input is less than about 112 volts).
For now, until you can move to a RV site with a good 30 amp plug in, I would leave the A/C off. Perhaps they can move you to another site, or the cord will reach to the neighbor's site - if they are using the 50 amp, then the 30 amp probably will be empty.
I carry a 50 to 30 amp adapter, because of the problem you described, I was at a campground with 30 and 50 amp plug ins. My volt meter indicated low voltage (108 I think) and it was 105 outside. I plugged in my voltage booster and 50 to 30 amp adapter (the cord end was very hot) and when I ran the A/C my voltage was then 125 volts. With the A/C off, it jumped to 132 volts,
Also your refrigerator on electric will add about 3.5 amps to the power consumed through your power cord. Changing it to gas will make it colder, and increase your RV voltage by about 1-2 volts. On electric, it is only about 900 Btu's while on gas it is 2,200 Btu's to the boiler that runs the refrigerator.
Good luck,
Fred.