Hvar or Korkula?

Gwen, I would say choose Korcula. I found it more beautiful and less touristy than Hvar.

Haven’t been to Hvar, but have been to Korčula numerous times. We are going again this summer. Korčula town is a nice size, you can walk around the town very quickly, meaning the circumferfence. It has a old town with a labyrinth of streets to explore. The Church has a small tower you can climb and there are a few small museums, including the house of Marco Polo. The food is good and not expensive.

By the end of September, both Hvar and Korcula will be relatively quiet with some restaurants about to shut for the winter. I don't think you will find Hvar too touristy at that time of the year. I'm assuming you mean Hvar town and Korcula town as options to stay. On balance I'd go for Hvar, not least because it claims to have the most number of days sunshine, but providing the weather stays good you won't be disappointed with either one.

For only one night, either would be suitable at that time of year, as places will be winding down for the winter closures at the end of September.

I was on Hvar island in early/mid September a couple of years ago and the catamaran was cancelled twice due to high winds and rough seas. The ferries were still running.

They are both pretty. If I have to choose just one - then Korcula (read Korchula).