Yep. Your "5000 watt" contractor generator only puts out about 2400 watts (20 amps) per 120VAC circuit, and some lower-priced units are overrated and may not do that in the real world. You can't combine the two 120VAC outputs into one because they're 180º out of phase.
In general, a real 2400W generator is enough to power an air-conditioned Airstream, at least within a couple of thousand feet of sea level, but other demands have to be minimized. For sure, you have to be running the refrigerator off gas rather than "auto" which uses 120VAC when available. It's best that the batteries are pretty well charged so the converter isn't drawing much from the generator. No microwave, hair dryer, etc while the AC is on.
For what it's worth, a 5000 watt contractor generator should have enough horsepower to maintain 2400 watts on one circuit at higher altitudes. Just don't be surprised if you're asked to shut it off or leave in many parks.