The beach towns in NJ all have a different feel. If you have younger kids you cant beat Ocean City NJ. We have a family house there and visit every year. Most other times it's rented, but the amusement rides and food on the boardwalk are great. If your kids are older, or just adults then Wild Wood or Sea Isle are good options. Ocean City is a dry town so very much family focused. The others are nice for families as well but have bars and clubs and more night life. Cap May at the very bottom I think is the best if your looking for almost that New England feel. Great town to walk around, shop, eat, etc. As stated before, the campgrounds are a mile or so from the beach, but some are nice. There are 3 or 4 within about a 5 mile stretch toward the top of sea isle which is basically right below Ocean city. We have stayed in this area and enjoyed it as well because you can go to both towns easily. Again, you will have to drive to the beach.
And in reply to the posts above for assateague, I agree on the beauty of that area however you have no chance of getting a site for this year in the state park. If you were planning for next year then you might have a chance. You could stay at Castaways or one of the private parks off the island however. To search this you would look in the Ocean City MD area. Good luck