I've made the trip twice and found very few campgrounds open until you're well into VA at that time of year. Either make reservations, or do the rest area/ Walmart parking lot thing. We did stay at a KOA off I-77 just south of Charlotte,NC. Don't know where you're headed in FL, but we also stayed at a KOA in Stark FL on RT 301.
In 2012 we left Oswego just after Christmas and got caught in winter storm "Euclid" in southern PA. Crawled along in 4 wheel drive at 10 mph for about 4 hours because I had reservations at a Campground south of Carlisle called Western Village. Woke up with 6" of heavy wet snow on the ground and trailer. Without 4 wheel drive I don't think I'd have gotten out of the park.
My other recommendation is to stay a long ways away from Baltimore, Washington DC and Richmond VA. We take I-81 South to I-77 below Roanoke, VA. I-77 to Charlotte, and I-26 through Columbia toward Charleston where we grudgingly finally join the I-95 rat race.
2015 Rockwood Ultra Lite 2604 WS. Rear LR, large street side slide with small wardrobe slide in BR.
2014 Ram 1500 Eco Diesel Laramie Quad Cab.