As you have heard before, damper is relative...and the more barren areas it grows in nature, have good drainage, usually a slope. And....damp in Arkansas or Oklahoma is a lot different than damp in the NE. I'd suggest well-drained in your area.

I like to collect various seed in the Fall and store in my unheated garage during the Winter. I baggie them individually c names, done c a Sharpie marker and store each baggie, inside an empty glass vase. I have not tried this c Mountain Mint but I'm sure it would work. You can also store seed in the fridge over Winter, if your spouse doesn't kick up a fuss (mine does LOL). You can also shop at specialty nurseries in your area that deal in natives to your region.

Looks like Nepeta cataria, Catnip.

You should be in good shape. I've never covered mine and it's doing quite well.