I've only been to Asheville once, but there's a spot downtown called Mela that has some of the best indian food that I have ever had in my life. Went there for dinner and they were super accommodating to making our meal sans rice, etc. and giving us a salad and veggies instead. Amazing flavors, huge portions. They have a gourmet whole foods-like supermarket near the downtown area (which, may I add, had a huge gluten free/celiacs special day when we were there- tons of vendors offering free samples). Of course gluten free doesn't equal paleo, but they had a nice selection of paleo friendly snacks.
If you are willing to go out of Asheville, there are lots of farms that were pick your own (we went to Leicester in July and got tons of organic blueberries for cheap). I imagine if you go there now, you may find squash and things.
During a trip to Asheville several years ago, my wife and I went to a place downtown called The Tupelo Honey Cafe. It's not "paleo" per se, but plenty of paleo friendly options and meals that could easily be modified to fit. It WAS 100% delicious though :)
Luella's is THE BEST BBQ I've ever had. And I'm a southern native. Their sauces are Gluten-Free as well.
Earthfare (West AVL off of I-240 coming into town) and Greenlife (north Asheville, Merrimon Avenue) both have pretty good paleo options. Local as well (though Greenlife was recently bought out by Whole Foods). I would recommend Greenlife over Earthfare for meat. The farmers' markets should still be going on throughout various parts of the city on different days.
If you're willing to put out $$, there's also the Fresh Market on the same street as Greenlife but a bit farther north. They tend to have fancier, "higher quality" food.
Good restaurants (somewhat paleo adaptable): Papa's & Beer (mexican), Tupelo Honey ("southern fusion", but just really damn good), Luella's, 12 Bones BBQ, Posana Cafe is all gluten free and has great paleo dishes, and Early Girl Cafe (local, "southern", good meat).