My rule of thumb is that we can travel roughly twice as long in the RV versus cost of our non-RV trips so far. As noted fuel and campsites are the biggies.
Variables on season and location. Groceries and fuel are higher in Canada's Atlantic Provinces, for example. East coast has very pricey areas. Around DC, NYC, prices are high for camping close to the city. You'd find the Carolinas fairly reasonable. SC has low fuel prices.
Resort prices kick in if you want to be on the beach or in the mountains during summer high season anywhere. Note you CAN get sites in great National and state parks in those areas but you'll want to plan ahead and book very early. Hunting Island SP, for example, is great, right on the ocean, and the most popular SP in SC.
I'd say your best timing would be late winter to early summer for a trip east. It's hot and humid in the rest of the Carolinas or anywhere on the east coast in the summer so might want electric sites for AC. Spring is very pretty here.