There aren't any campgrounds close to downtown, but Tom Johnson isn't too far away. However, it could be zooy if there is a race scheduled while you're there. There is no major NASCAR race there in March, but the speedway also has a dirt track and a drag strip. You should be OK that early in the season, but check the track schedule just to be sure before booking the campground.
We usually stay at Carowinds when we're in that area. It's just over the border in SC.
There is also a KOA
just over the border in Fort Mill, SC that is open all year. Don't know much about it.