Looking for a charming small town between Copenhagen and Stockholm in late August

I haven't been there before, so I can't recommend, but we are traveling by car from Stockholm (4 nights) to Gotland (3 nights) to Kalmar (2 nights) to Ystad (2 nights). From what I have read, both Kalmar and Ystad seem to fit your description.

Are you limited to train or would other public transit options work?

I really enjoyed Visby on the island of Gotland, but it is not exactly on the route to Copenhagen. It is an easy round trip from Stovkholm with a trip on the pendeltåg (commuter train, covered by SL card) and then a ferry to the island.

Kalmar is an obvious choice, but I actually preferred visiting nearby Öland which I did with a car.

I haven't been to Ystad or Lund, but either could fit your bill. Ystad is the setting for all the Kurt Wallander crime novels.

The town of Mariefred is a great day trip from Stockholm ... nice town and the wonderful Gripsholm Castle.

While driving in 1994, I did enjoy an overnight at one of the "kopings" (north, south, or new? I forget). But if traveling by train, I would not break up the 5-hour direct train ride from Copenhagen to Stockholm. You can get your small-town variety in towns that Rick lists for excursions from those cities, but which don't happen to be along that direct train line. I'm also a sucker for the architectural history on view in open-air folk museums, but they are not "living" communities.

I think I mentioned this in an IM. Here's the visit Sweden info:
"Eksjö lies in the county of Småland, in southern Sweden. Eksjö is rightly famous for its wooden buildings, many from the 17th century. The old part of town, with its shops, churches, squares and houses, is straight out of a Pippi Longstocking movie, with its copper red, cream and citrus-fruit coloured wooden buildings. The cobblestone streets and alleys add to the historical flavour of Eksjö and you really should go on a guided tour with one of the locals and visit the town museum. A visit to Eksjö is about enjoying the simple things in life; rent a bicycle and discover the town and surrounding countryside with its red farmhouse cottages dotting the landscape, go hiking in the nearby forests, or laze away a day beside one of many lakes here. A great idea is to rent a car and discover the world famous crafts and furniture design of the region."

"While you’re here, visit Skurugata, a ravine hewn out of the landscape, just outside Eksjö."

It is a charming and warm town and fits all your criteria. Stay right in old town. Exit the train at Nassjö. Buses leave the train station for Eksjö. Easy.