[Quanah Parker]

I'd like to toss Hiawassee, Georgia into the mix.
I'm a fan of northern New Mexico and the Ozarks, too.

I'm sure you realise there's tons of rural in the US, but you've set forth some hard to match criteria. You realise the entire southeast, Apps and all, have tons of humidity? You realise most rural areas have plenty of Republicians? Getting out in the country doesn't automatically mean you'll be left alone. Fewer folks mean strangers attract more attention at the store, etc., and with less going on, any stranger is reason for excitement and the locals will notice you. But maybe you mean rural, as in, 10 or 20 acre plots where lots of folks have space, and vacation homes? Hiawassee fits that description with lots of folks from Atlanta buying vacation homes there. Awesome lake(s) and maybe 10-20 miles from the Appalachian Trail and National Forests.