We have traveled in our 24 foot Class C RV from California to the East Coast and Kentucky a couple of times. Once through the Southern U.S. and once through the Central U.S. ... about 9000 miles on the first trip and 7000 miles on the second trip.
For specific reasons we had no control over, both trips were during July/August/September .... and that was the only downside of the two trips .... the extreme heat-humidity mix that time if the year in those parts of the country. I suggest that your RV not only have excellent air conditioning for both hookup and drycamp situations, but also a good dehumidifer in it - to be on the safe side comfort-wise. We had to run the air conditioning with the built-in generator almost all night once in order to be able to sleep.
Good luck on your trip - you'll especially love Death Valley and other Southwest areas if you can keep cool enough that time of the year. We're planning on another trip this Augus/September/October to the East Coast - but through the Northern U.S. this time.