My calendar shows July 4th to be on a Tuesday this year. The 5th, 6th and 7th are Wednesday, Thursday and Friday.
We left Yellowstone on July 2nd last year and arrived home in NJ on the 11th. We stayed in private campgrounds all but one night (which we spent in a Forest Service campground) and had no problems getting a site each night or issues with having to stay or pay for more than one day.
July 4th is a Thursday in 2013 and the OP has a valid concern as that Fri-Sun will be considered a holiday weekend.
They may have better luck with a reservation as we get 30 days or closer.
To the OP, I feel your pain. I only have two children, but I remember vividly the first time we had to buy them seats on an airplane and how it doubled our cost of flying.
Well this added nothing to the conversation. It's not funny and the comparison is way off.
This is my angle, the OP may have been onto the same thing. I find a campground I am interested in, either through a friend or mentioned here. I go to the website and click rates. It is not always easy to find the true rate you have to pay. Some just list a flat rate. Some list a rate and then you see some small print that the rate is based on 2 adults and each child is an additional $ amount. Bayley's in Maine for instance. A lot of people recommend it. You go to the rate page and look how far down you'd have to go to see this rate is for 2 adults. People not familiar with this pricing structure may not even think to scroll that far down. The fact that the rate is for 2 adults and I'd have to pay an additional $18 a night for 3 kids is truly buried at the bottom. Be upfront!
National Seashore RV Park would be the best choice for you. It has a water park and a lot of kid activities. There is also a Zoo near the Park. So I'm sure your kids can have lot of fun over there.
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Where is this place? Google search of National Seashore RV Park comes up with many possibilities.