There are Self Service campgrounds, Backcountry areas, and dispersed camping.
THIS PAGE gives more information on all of them. There are also pictures.
Many of these self-service and back country areas are close to the water, and have been known to flood. There is a visitor's center at both the north and south entrances, and also near the middle, so you can stop and ask.
In your shoes, I would go to one of the developed campgrounds first. On a non-crowded weekend, you should have plenty of space. Piney is the largest and also has some of the nicest sites near the water.
ENERGY LAKE might suit you it is a smaller campground on a calm lake. The campground is not the most developed, which you also might like. But the selection of sites for a large trailer is limited.
Just so you understand.... the lakes of LBL are impoundments of the Tennessee and Cumberland Rivers. They are large navigable waterways with heavy barge traffic. I would not go near the main channel in a kayak.
The sites right on the big lakes are going to be windy and possibly not pleasant on a cool spring day. Getting a little off the water is not a bad thing. Keep that in mind.
You can get a detailed map
Single empty-nester in Middle TN, sometimes with a friend or grandchild on board