you can search on www.woodalls.com for campgrounds that are open all year. Search by city first, then filter.
It would help if we knew how far up the coast you want to go. It would also help if you told us what you like to see or do.
I would cut over on I-26 toward Charlotte NC, then take I-81 east. Stay on I-81 to Scranton PA. Beautiful drive. We like to stop in Mt Airy NC at Mayberry Campground. They are open all year. We drove to Pilot Mountain State Park and drove up to the top. Not sure if it will be open then. Stop in WV at Harpers Ferry. We stayed at the KOA. They are open all year.
If you get to Pennsylvania and haven't been to Gettysburg that would be a good stop. Also Niagara Falls before you head south.
In Nashville we like Two Rivers Campground or Seven Points Corps of Engineers campround out of town.
To miss Atlanta, head south from Nashville through Birmingham. Or if you have time, take the Natchez Trace Parkway to Natchez MS, then head back east.