I'll add to the it-depends-on-where-you-are responses. I live in East Nashville. The norm here is "really liberal." You'd probably feel like an outsider in our neighborhood if you were a really conservative white guy with a Trump sign in his front yard. A liberal feminist bi-racial woman wouldn't even show up on anyone's prejudicial radar. Outside of Nashville, it's often a different story, but even then, not necessarily.
I can't speak to the racial component very well, since I'm just an average white guy, but you can find liberal enclaves throughout the South if you do a little homework. Nashville, Atlanta, Knoxville (although a very white town in general), Chattanooga, Asheville (which, by the way, also has a very, very conservative element), Greenville SC, North Charleston SC, Raleigh-Durham NC, South Florida, Austin TX, Athens GA. Outside of Nashville, there's a town called Woodbury (pop. 2,600) where some people, many of them gay liberals, from East Nashville are moving to escape the growth here. As far as I know, that's going ok for them. Those are just a few places I know off the top of my head. I'm sure there are many more like them.
If you go to these places assuming that you'll be seen as an outsider (not saying that you, personally, are), you'll probably find evidence of that pretty easily, but I don't think the night-and-day difference between North and South that many people imagine still really exists. When I lived in New York City for several years, I saw plenty of examples of racism, religious prejudice, and closed-mindedness, including the cop who told me without hesitation that I should be careful about moving to a particular neighborhood because of the high school there, where they "bus the n****** in from all over the city."
All that to say, if you end up deciding that you want to move south, I'm pretty sure you could find a place where you could feel at home. Good luck! Hope things work out well for you in any case.