If you're thinking of coming as far west as the Pensacola area, then I'd suggest Fort Pickens CG in the Gulf Islands National Seashore. There are 3 camping areas, with various size sites. All the improved sites have water and electric, picnic table, and grill. There's a dump station and bath houses. None of the campsites are on the Gulf, but some are within walking distance. Others are almost within sight of the Sound. Cost is $20/per night. Since it's just minutes from our home, we took our RV there for our first outing so if we forgot anything vital, we could get it easily.
Just be aware that the road from Pensacola Beach to the fort and CG is basically at sea level in a few spots. The dunes have never recovered from the damage from hurricane Ivan in 2004. So when the Gulf gets really rough, with a strong wind pushing the water toward shore, that road will sometimes flood. Just this weekend, they evacuated the campgrounds out of fear that TS Debbie would flood the road and campers would be stranded. Not a frequent occurrence, but something to consider when planning.
If you don't want to come so far West, we stayed at St. George Island State Park near Appalachicola and liked it. Again, sites aren't on the beach, but it's easy to get to.
These are the only two CGs we've stayed at in the Panhandle since we live at Pensacola Beach, but there are others that get good reviews. Try searching here and I'm sure you'll find more suggestions.
Fran & Tom/ 4 rescue cats - Benci (2014), Peaches (2003), Tipper (2002), Snippet (2002), & in spirit - Aja (1996-2014)/2011 Winnebago Vista 30W/2008 4 door Wrangler 4X4