I've traveled up and down the east coast in all four seasons and engaged in blacktop overnights. It can be convenient, particularly when campgrounds are closed. But, JMHO, I think you'll generally find that Sept. in the S.E. is still uncomfortably hot and humid at night sans air-conditioning.
You could mitigate the expense of RV parks by using Passport America or government campgrounds along the I-81 corridor, assuming that's your route . But, that's going to involve more planning/scheduling than just stopping at a Wally World or Cracker Barrel.