re: Shortest trip to a "quaint little mountain town" ?

Chestnut Bay resort in Leesburg, Alabama. You can rent a home on Weiss Lake. Tons for the family to do as well. Roughly a 7 hour drive.
Desoto Falls
https://www.alapark.com/desoto-state-park
and the Little River Canyon
https://www.nps.gov/liri/planyourvisit/little-river-falls.htm
are nearby.
https://chesnutbayresort.com/
I've stayed in house #5 and they allowed pets. There are two pools and two waterslides.

Another option is Gorhams Bluff. I believe it's located in Pisgah Alabama. It is a large house on a bluff overlooking the Tennessee River. It is supposed to be like an artist colony based on Seascape in Florida. You can rent freestanding cottages or stay in the main house. Here is its link:
https://gorhamsbluff.com/welcome/

Lastly is a place called the "Secret Bed and Breakfast" in Leesville, Alabama. It is on a bluff overlooking Weiss Lake. Of the three, it would be my least favorite, but still okay.
https://www.secretbedandbreakfastlodge.com/

All three are located in the same area. The museum for the country band Alabama is not too far away in Fort Payne as well. I haven't done that. As for which of the three above to recommend, I would suggest Chestnut Bay. The view from house #5 is legit. I think we also stayed in house #32 (The Who Dat House). I preferred #5. I hope this helps. Enjoy.