Driving down the road of course should not make any difference than sitting still with the genny running and ACs on.
The only thing that comes to mind in this situation is a loose 12v control conection affected by the movement/vibration of driving.
I'm not familar with carrier systems, but they should be similar to dometic systems. Which use standard telephone wire and the standard RJ-11 jacks to control the system. Checking/cleaning/tightening the connections at the Tstat, Zone1 and Zone2 ACs would be my first move. Let us know what Carrier/Monaco has to say.
John & Clare - Buck & Zeus - Jack & Husky
2008 Fleetwood Bounder 38P
Towd: 2 HD Streetglides