When we first got our MH (used)I asked Fleetwood about running the wash machine while going down the road and they said that was fine, but I couldn't run the dryer without starting the generator. From that I deduced that the washer didn't draw a lot of power but the dryer did.
So yes you can run the washer with the generator but it probably won't put much of load on it so run the dryer also for a real load on the generator.
We normally don't use the dryer. Instead when we're parked with both slides out we use a clothes drying rack.
Bill