Middle of March is a crap shoot in VIrginia. You could have anything from snow to 80 degrees and sunny.
In general, the tidewater is going to be warmest, and the mountains to the west are coldest. I remember wearing boots to go from southwest Virginia to an interview in the piedmont, only to be embarassed when there wasn't snow on the ground when I arrived (and the boots were the only footwear I'd brought, because I'd seen nothing but snow for two months).
The further south you go, the better off you'll be as far as weather. There's a good chance the weather will be pretty good in the southeastern quadrant of North Carolina, roughly Morehead City and south. The coast of the Gulf of Mexico may be a 1.5-2 day drive from Cleveland, but it's a good bet for decent weather.