John, welcome to the forum!
It helps if you also post what type of rig you have. A 21' coach will fit in more places than a 45' one and your responses will be much better.
Camp Gulf is a pretty intense park. Do you intend to stay the entire summer.?..June-August? of just a week or so. Gulf has sites right on the beach..a bit pricey and lots of sand tracked into your rig. It's one of the few along the Emerald Coast where you're right on the beach. The sites are not spacious but the location makes up for it. The rest of the park is a pretty typical RV park.
Henderson State Beach is IMHO, the best state park in the area but it does not have sewer hookups. With GKs that might be important. Their restrooms are spotless though. It's a short walk across the dunes, (raised wooden pier) so no camping right on the water due to the dunes being protected. Very nice beach and pretty quiet. Easy swimming. It's close to WalMart and a free alligator farm is across the highway. There's also the Track for waterfun and go carts.
I have pictures of the area, including the gator farm, on my blog...link below.
For a full hookup State Park, Topsail is just east of Camp Gulf. Nice sized lots and they have a swimming pool. They're a bit off the beach but offer a shuttle service to get you there. While we found Henderson to be a cleaner park, the sewer hookup might sway you to stay here instead. Also, with the kids, the pool might be a big deal. It's a bit out of Destin so shopping etc. is a ways.
West of Destin is Fort Walton Beach and Destin West RV Resort. It's a restricted access resort, Class A, C and fivers only. Like Camp Gulf, expect to pay about $100/night. It's on the bay not the Gulf but it has very nice facilities.
Again, welcome to the forum...hope this helps....Dennis