Most, if not all states have a state park web site. I have no experience at Federal parks as I’ve visited many , but don’t stay at them. Reservations and what’s available can be found at the state park web site. Your success in reserving a camp site will often depend on where, what time of year and even days of the week. Popular destination often require making reservations many months in advance depending their reservation system.
To each his own, but we prefer private RV parks near an attraction, rather than staying at the attractions campground itself.
good luck and safe travels.