I HATE camping. How old are your kids? My top budget choice is Outer Banks the week of Labor Day if your kids are too young for school. Rates drop WAY low, you can drive and shop at the cheap southern grocery stores. Plus cheap beer at Brew Thru.
Local, close to here, I would recommend Cape Charles, VA. You can rent a big house and split it with other low budget friends for cheap. You can also get a small house for a family of 5 or smaller like this for $700/wk in Cape Charles.
https://www.vrbo.com/3804905ha#rates
Another local option would be renting a house on the Eastern Shore of Maryland that has a bayside beach.
Deep Creek Lake also has some reasonable rentals, and Lake Anna.
Definitely a rental cottage. Pick one with average decor, it will be cheaper. Maybe settle on a house with only one bathroom and an outdoor shower.