We have done a Christmas/New Years trip from N.E PA. to Disney, for the last dozen years. We have never had an issue finding an open campground, and some go out of their way to provide easy after hours registration. We have stayed at Green Acres in Walterboro SC, many times, but have never spoken to any staff there, or set foot in the office. Giant pull thrus, and a nice well lit open shed to self register at, after hours.
One important thing to remember with a trip like this is that a open campground, a quarter mile from an exit is a good thing. An open campground that is a fifteen or twenty minute drive off the exit, while making 1/2 dozen turns, is something we work hard to avoid while in transit to a destination, especially after dark.