4 nights on the islands Hvar vs Korcula vs ????

Pick one--flipping a coin is not a bad idea (seriously). You'll be doing the same thing no matter which island you pick--enjoying the stunning scenery and enjoying the outdoors, hopefully taking a swim. So, I feel like doing that in a leisurely manner is far preferable to squeezing in two stops.
What month would make a difference. I would probably avoid Hvar town in late July/August--though you could pick Stari Grad instead. It just always had the reputation as a party place in high season (I only visited in May and it was chill then, but it's been several years). Any island will be more crowded in July/August.
There are boat trips on all islands, but we really enjoyed renting a scooter on Vis to get all over the island. If you are very fit, you could also rent bikes (the hill out of town is pretty step).

We stayed 3 nights outside Hvar town on Amfora beach and 2 nights in Korcula Town. The 3 nights allowed time to hike up to Hvar’s castle ruins one day and take the bus to Stari Grad— one of the oldest continuously-inhabited towns on earth. ( I’m a history/archaeology fanatic). We saw archaeologists unearthing Roman mosaics under Stari Grad’s sidewalks the day we were there!
You will not run out of things to do on Hvar because the island is quite large with other interesting towns and beaches to visit.
Korcula Town is a really amazing place— but very small. The architecture looks like Jerusalem with stone blocks used in the construction of the buildings, ramparts, gates and parapets. We walked around Korcula Town’s perimeter in about 30 minutes and then explored the town and swam off the stone stairways leading to the sea water— which is some of the purest sea water on earth.
After one day, we took the 20-minute ferry to Orebic and swam at the beach before hiking a few miles to the Mikulic winery south of town. This is the Peljesac Peninsula, known for its red wines.
So, I would say that Hvar offers more to see and do than Korcula does—making it worth more time. In retrospect, I could have stayed just one night in Korcula Town and been happy with that, but we try to avoid one-night stays.
Both are very worthwhile.
I ran across an article yesterday which interviewed 7 travel industry experts who all agreed they view the Dalmatian coast as the best area in Europe to visit. Those interviewed specifically named Hvar and Korcula as being the fastest-growing hot spots of European travel in the very near future.
So, if you can visit both— now is the time to do it.
We all know what Dubrovnik is like.
Carpe Diem!

I enjoyed Korcula town more than Hvar, it just spoke to me. Still two nights would be my choice for either if I were out to see the region over a 2 or 3 week period … maybe more someday just as an escape. There are several good wineries nearby and another unexpected interest that has been repeated over the years is stopping at places that produce olive oil. Again, several to choose from.

If you trip is early September, maybe some beaches; if its late September maybe not some beaches. But I love the region in September.

Kenko loves his history and archaeology so for him and you here are a few lists of the oldest towns in Europe. Sort fascinating which cities are on the lists; a few I didn’t expect.
Top 10: https://www.worldatlas.com/cities/the-oldest-cities-in-europe.html (this one has good descriptions).
Top 20: https://blog.tours4fun.com/discover-the-20-oldest-cities-in-europe-history-journey.html
Top 30: https://www.farandwide.com/s/oldest-cities-europe-c9848a0d07ab4c3b
an even longer list: https://www.eupedia.com/history/list_of_oldest__prehistoric_european_cities.shtml

I read a couple of articles the other day that mentioned that tourism in Croatia is down 16%. One attributed it directly to the fact that tourism costs have increased by as much as 50%. This one discussed it a bit more and might be more accurate: https://www.travelandtourworld.com/news/article/croatia-tourism-faces-setback-foreign-arrivals-drops-16/ I sort of hope its true. My first trip a dozen years ago was great, my third trip about 3 years ago places were tourist zoos..... But if the trend continues, it might be become better for those that can still afford it.

My gut reaction is to just pick one four night stay. Two nights each kind of shortchanges both, and I wouldn’t bother with one nights. (You also might find booking less than three nights gets difficult.) Especially given all your interests, you’ll want some time to find your flow.

I think you’ll be very happy on whichever island you land. This area is just stunning, seemingly everywhere. A quick scout of accommodation options may help tip the scales for you.

Both islands are very nice and I'm sure you'd enjoy either. If it were me I'd pick Korcula- the old town is amazing for just wandering around at night. If you like outdoorsy activities you can take the ferry for a day trip to nearby Mljet which is perfect for cycling, walking, and kayaking. For beaches, you have Lumbarda which you can get to on a small ferry from Korcula Town. I seem to remember a vineyard nearby as well. I found Hvar Town very crowded and more oriented towards nighttime revellers, tho Stari Grad was peaceful. If you're looking for beaches near Hvar you can take a small ferry to the Pakleni Islands which have lovely (and quiet) beaches. Personally when we went we did three nights in Hvar and two in Korcula. Good luck deciding.