RnR, as a North Carolinian, I will comment. I totally, completely get what you are saying. I was born and raised in Raleigh, the state capital, a university town, and much more progressive than either western or eastern NC. There are a few well-known "blue" enclaves in NC, and these tend to be much more tolerant of all forms of diversity (in religion, sexuality, etc. etc.) (I am also not religious.) I always thought Asheville, in the western part of the state, and probably where you are looking, would be included on this list, but according to this site: https://www.roadsnacks.net/these-are...orth-carolina/, it is not. However, there is a strong artistic community there, and it is always thought of as a diverse and progressive city.
I have lived in each of the four top cities on the list from the site above, and I can vouch for them. They also have better cycling than the western part of the state, which is mountainous, roads are skinny, most have no shoulders, and very short sight lines. It's absolutely beautiful, but I don't feel comfortable riding in most places in western NC. We own a small piece of property in the NW corner of the state, on a river, and I would not ride my road bike there. I only ride my MTB on the dirt road the property is on, which is very rural and a dead end road.
The cycling in the Piedmont is much better, though you will still run into the occasional problem with loose dogs, rude drivers (who tend to be driving the huge pickup trucks), etc. The rural areas of the Piedmont are a lot more conservative than the cities themselves.
So, you might want to check out Chapel Hill, Carrboro, Hillsborough, and Durham. I am partial to Hillsborough, as it is the smallest of these cities, but they all have their good and bad points. Durham, to me, would be the least cycling friendly of the four. Chapel Hill and Carrboro have the university, some bike lanes and paths, a good farmer's market, lots and lots of crunchy granola types, runners, cyclists, outdoor activities. I love this area and would certainly live there again if the opportunity arises. Chapel Hill is the most expensive of these cities but probably very decent by your northeastern standards.
I know you are a trail runner, and I think you would find places to run in all of these communities. I suspect Umstead Park (which is in Raleigh) is the nicest park for trail running (I've MTB'd there but not run), but I am sure there are many others I am not even aware of.
If you have any questions I can help with, feel free to ask!
Emily
2011 Jamis Dakar XC "Toto" - Selle Italia Ldy Gel Flow
2007 Trek Pilot 5.0 WSD "Gloria" - Selle Italia Diva Gel Flow
2004 Bike Friday Petite Pocket Crusoe - Selle Italia Diva Gel Flow