Yes, you need 12v to power the control board in the fridge regardless if running on 120v or gas. As long as the converter/charger in the trailer is putting out 12v you shouldn't need a battery.
However, some older converter/chargers won't put out 12v without a battery in the circuit. If this is true for yours, you won't have any 12v power at all. If you other 12v items working like the water pump, interior lights, and radio, then you have 12v in the trailer, and the fridge issue is likely either a fuse of bad control board.
There are tons of YouTube videos on diagnosing an absorption fridge...you want to diagnose well because it gets expensive in a hurry if you just start throwing parts at them.
