Enjoyable or miserable? Depends on how you deal with hot and humid. Whether or not you do it in a RV, you will find out about early summer outdoor life in that part of the country. June is a pretty good time to go. July and August days don't get much hotter, they just have more hot days and fewer cool days as the Gulf Stream moves in toward shore, and by August we would just be getting tired of it.
Although from Southern Michigan where "hot" is anything over 80F and humid is a regular thing, I've lived much for the past 45 years in places that get just as humid, some that get a little bit warmer and stay warm longer (Coastal Carolina, Central Florida) and places that get a whole lot hotter (highs over 100F for two to three months) and are considered humid when a 70F to 80F dewpoint puts a floor under night temperatures in the summer.
I go further south into Texas and the Gulf Coast for summer RVing. I think the RV can be kept comfortable if it can be parked in shade and the air conditioner kept running full time (that's why my motorhome has its own generator). But my frame of reference is that I keep my own house no more than 20-25 F cooler than outside temperatures, thus 80-85 inside most of the summer. If I can pull the inside of the motorhome down to 90 F, I'm happy, because the air blowing from the vents is coming out a cool 70 or so.
For you, that might be hot. You might not want to ever go outside.
The hottest part of the trip will be driving across the Great Plains in June. If you want to interest your wife in full-time RVing, I would not take her across the plains in a RV at that time of year.
The most humid part of the trip will be going across the Middle South to get to North Carolina, as hot wet Gulf air masses will already be pushing as far north as the Great Lakes states. The Piedmont will be getting pretty warm in June, the Atlantic coast will usually be a little cooler with air blowing in from the ocean, but quite humid. In the good old days the folks who owned the tobacco plantations on the coastal plains would already be in their mountain retreats by June (that's why their school year ends in early to middle May). Those who stayed behind would be sitting in the shade waiting for harvest.
Your NC destinations are mostly in the Piedmont and lower part of the mountains, not as muggy as the coast, but still a long hot summer. Piedmont elevations in the Piedmont are not high enough to get any useful adiabatic cooling effect. Wilmington gets cooled by the sea air, warmed by air blowing off the Piedmont; at least it is not too far from the beach.
Savannah and Charleston are best appreciated in autumn and early winter, I found them oppressive in the summer, but that was also the first place I lived and worked after leaving Michigan, and cars back then did not have air conditioning unless they were luxury cars sold in Texas or the Deep South.
There are two sides to a mild climate, not too hot and not too cold. East of your Pacific coastal range, most places that do not get long, cold winters get long, hot summers instead. That would be the Atlantic Coast. All the rest of the country most places that don't get below freezing and snowy gets long hot summers; in the middle, even the places that get long cold winters will get quite hot summers.
