Do any of the 120V appliances work while plugged in? Outlets and microwave are really the only thing that doesn't need 12v power. In the case of the A/C it is probably just the thermostat that needs 12v so when the 12v power goes the thermostat is probably shutting down the A/C.
I agree with Gomez. Check to see if your converter is even charging your battery. Multimeter is preferred but even just the push button "battery gauge" should give you an idea. It should read full anytime your plugged in to shore power.
2014 Jayco Swift 281BHS, 300W Solar!
2015 F250 XLT 4x4 Crew Cab, Short box, 6.2 gas