I can speak for HM/Cottonwood TH campsites (they are both in the same location) and they are first come first serve basis. I've been there a number of times and have seen the lot almost full with vehicles once during the peak season but there were plenty of sites open. I've also seen plenty of folks sleeping in their vehicles as well, which I've heard is free but don't quote me on that one. There's a $6 fee and a one night limit - pretty basic and seems to be the honor system because in my times there I have never seen a ranger or camp host checking. If it was me I wouldn't worry about getting a site or having a backup.
