I friend of mine had AC issues also, had to be plugged in to AC power to work but the AC wasn't good. Found two issues, the transfer switch was bad so he had low voltage on gen power and the AC unit's compressor was bad also. Based on it's age he replaced the whole unit.
Not cooling means the compressor isn't running, but needing the genset instead of shore power points to other (electrical) problems. Need to tackle those separately, I think. Does the compressor start and run when using the genset? But still no cooling? What happenes when using shore power instead of genset? Does other 120v stuff work on shore power?
One possibility for the compressor not starting is a failed hard start capacitor. Easy and not expensive to replace. If you hear the compressor "grunt" and try to start but never actually get going, the capacitor is a good bet.
I would first thoroughly clean the condenser and evaporator and replace he filter or filters. When we picked up our 2003 Adventurer 38G, the AC blew ambient air. After cleaning the condenser and evaporator and replacing the filter, it cooled as it should.