Topsail Hill in Destin is a great campground. The beach is about a 1/2 mile or so from the campground. You can take the tram to the beach or ride bikes (we've walked it too). Very nice campground (full hookups and even cable TV).
Fort Pickens, Loop A, is very close to the beach. You just walk across Fort Pickens Road, down the boardwalk and you're there. As someone stated above, the campground isn't the nicest (they do not have full hookups, but do have a dumping station), but the proximity to the beach is nice.
One word of warning about Fort Pickens. The road that leads to the campground, Fort Pickens Road, is prone to flooding and, therefore, closure. Last year we went to Fort Pickens, but we had to evacuate due to Topical Storm Debbie. Fort Pickens Road was closed for a few days due to sand that washed up on the road.