You may well indeed be able to run the A/C once you have been on site for a few hours (Time to get battery topped off if it needs it) but NOTHING ELSE.
General rule on 20 amps is one of the following
A/C
Water heater
Microwave
Converter IF THE BATTERIES ARE LOW.. (forget it if they are full up) NOTE that this one is make/model of converter dependent, on a trailer, odds are it won't matter)
Yes, you will need an adapter, The campground may sell 'em, so does Wal*mart and every RV store from here to there and back.