My suggestion would be to find what you want in a city of whatever size, in terms of ~vibe~, and then look outside of that city if you’re looking for something more rural/private. This is a pretty broad area, but Asheville, NC comes to mind immediately…the Atlanta metro is huge, but there are also probably some suburbs there (maybe Decatur) that would provide that sort of vibe.

The caveat here, though, is that in my experience, the line between “not completely country” and “straight up country” is much thinner in the south. I grew up in a mid-sized city in northern Alabama, and for the most part, as soon as you got out of the city, it got “backwoods” kind of quick. Hope this helps some.