I split my time between public and commercial parks. While I prefer public campgrounds for their natural ambiance and larger sites, commercial parks offer more amenities at cheaper rates when staying for a month. Some public campgrounds will allow a 30 day stay, but I've found it is rare to get more than two weeks.
Summer is a good time for me to sit and visit the grandkids. Unfortunate in one regard, as N. Florida does get hot. Fortunate in another as I'm sitting for a month at a time and not competing for weekend space at many campgrounds. I'll float between 3 parks; two private and one city park which offers FHUs and allows a monthly stay. Each has it's advantages and disadvantages while providing me with some diversity and all are within close proximity to family members.
Once school starts, my travel days are generally Mon and/or Tue. I usually don't have a problem securing a site for a week or two as a walk-up. Did I mention that I don't make reservations? The "geezer pass" provides me with 1/2 price camping in federal parks and my home State offers the same to it's seniors at age 65. Popular area are best visited during their shoulder season.
If public campgrounds are your fare, there are numerous C.O.E. parks along the Savannah River and it's big lakes from Thomson, GA to Pendleton, SC. Many are open spring, summer, and fall; a couple year 'round. You'll also find some SPs on the GA and SC sides of the river that offer reasonable camping rates and a little diversity in scenery. With the "geezer" pass, the overall costs can be mitigated by alternating between the two if desired. Also, you might look around the Wilkesboro, NC area along W. Kerr Scott Dam & Reservoir. Several COE parks are waiting for you. If you like bluegrass music, MerleFest might be of interest. Heading east along the NC/VA line near I-85 you'll find more COEs and SPs at Kerr Reservoir. Then, there are lots of City and County parks that offer their particular charm in between.
US Campgrounds/info will assist you in locating many across the country.
Having driven some 800,000+ business miles over the past 30 years, I'm constantly amazed at what I've missed in between as I travel some of the same routes today. I suspect that you will slow down,
recognizing that as part of the journey, there's a lot to take in and experience along the way. Your destination will always be there when you arrive. Travel safe.