We have stayed at Henderson Beach many times. Love the privacy of each campsite. It is a quarter mile walk to the private beach or you can drive and park at the public beaches located at this state park. Shower houses are much nicer than some homes I have been in. Location is great to get to restaurants and shopping. Did I mention that Walmart is right across the road from the park entrance.
We have camped at Henderson and it is nice. Another option is just a few miles east of Henderson is Topsail Hill Perserve State Park. It has full hook ups there where Henderson only has water and electric, and I think Topsail has cable too. Like Henderson, Topsail is only about 1/4 mile from the beach and they have a shuttle service to the beach. Also Topsail has a pool and their sites have concrete pads.
If you are not set on Destin but just want to go to the Gulf Coast, I HIGHLY recommend the Gulf State Park in Gulf Shores, Alabama. All paved sites with full hook ups and quite inexpensive. We have been there many times and love it.