Our best pick for a peaceful stay near Branson is the Old 86 Corp of Engineers campground. It is located on a peninsula that sticks out in the lake, so most of the campsites are directly on the water and have beautiful views. The park flooded a few years ago and was overhauled, so the sites are wonderful, perfectly level with nice picnic tables. You can wade in the water from many of them, and there's also a swim beach.
But, I'm sorry to say it is not a FHU. They have water and electric, I do believe, and a dump station. It's really cheap. It is away from the traffic of Branson and very peaceful, but it is not super convenient to Branson, if you want to use it as a base for the tourist-y stuff. I know there are a lot of great parks around there, but we haven't stayed at any others.
You might Google campgrounds in Blue Eye since that's the southern part of the lake, away from crowds. A lot of locals also prefer Bull Shoals lake for less crowds. Lake of the Ozarks probably doesn't have the traffic problems of Branson, but it is a party lake, so you'll encounter the crowds on the water. They have done a lot to improve the roads around Branson, and there are a lot of side routes now that are less traffic.
