You can check availability online. https://northcarolinastateparks.reserveamerica.com/camping/Hanging-Rock-State-Park/r/campsiteCalendar.do?page=calendar&search=site&contractCode=NC&parkId=552854
I do believe they require you to call because not all sites are open and with the covid they would like for people to call ahead and make reservations. If you go on thier website you will see all the contact info and hours of operation as well as to what they are allowing.
I don't think they have camping at the cave side. You need to go to the website for hanging rock and check the reservation page.
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