There are some excelent suggestions already. I'm not too sure how far you want to drive, and that could influence this next suggestion of mine. Over here in Western KY, Union County I believe, you will find exactly what you are describing.... Mauzey Lake.

I've not been there since I was a child, but my sister took her family a few years ago and said it was pretty much unchanged. Best I remember they do have a campground too.

I tell you, there are a lot of places in this great STate that will fill the bill you are describing.

Danny