I think the club you're referring to is http://www.boondockerswelcome.com
. But, read the details- it's likely supposed to be just for an overnight.
As you mention leaving the camper, the best place for your setup is in a RV park or campground. I would never leave my camper unhitched at a truck stop or public place (Walmart, etc.). That's not what those places are for and I just don't feel comfortable with the security of them.
When I'm just picking a place to park while I work- my requirements are very low. In fact, I'll pick just about any place that reviews indicate it's "big rig friendly". Give me quality electric, water, and (maybe) sewer and I'm set. It's not a destination and I don't need to like the campground.
You've named off a bunch of big areas. We need more details as to what you want to be near.
For "Eastern PA" - I stayed outside of Pittsburgh at Fox Den Acres. Level stone lots, but not much to look at. They did lose water after losing power in a storm (well/pump, I assume).
Baltimore/DC was my hometown. Whispering Pines in Woodstock is a good middle point, or Cherry Hill in Silver Spring, or Kapitol KOA in Millersville, MD.
In Richmond, it's a crap-hole, but Kosmos RV Village was decent for an overnight stop on our way south. Pull thru-lots and inexpensive (so long as there isn't a race somewhere nearby).
Atlanta, the only place I've stayed at was Stone Mountain Park Campground. Decent place, but expensive at like $60/night.
Miami, I stayed at Larry and Penny Thompson park. Great place, if you can get in. It gets crowded/full in the winter.