Cape May or Wildwood, NJ

There dozens of campgrounds in Cape May County - from plain to fancy, but none of them are cheap, and they all are going to be busy in August, but the later in August the better, as kids go back to school earlier and earlier. Older kids will want to hang out at a boardwalk - Ocean City is most family friendly, but Wildwood much more fun. You can take dogs on the beaches of Lower Township, but they are on the Delaware Bay side. The only ocean beach I know of is Stone Harbor where you can take them between 7 PM and dark, but they have to be leashed.
Cape May County Campground Association
www.jerseycapecamping.com