Best Experiences Kotor, Montenegro, Dubrovnik, Hvar, Opatija Croatia

I would have to know what the ship was offering to advise whether independently was better, but these are not large places--a quick perusal of a guide book or web guide would likely tell you what to do. In short:
Kotor--see old town, hike up above it if inclined OR do a bay tour (both if time allows)
Dubrovnik--walk the walls, walk old town
Split--Diocletian's palace
Opatija--walk the lungomare, gawk at the villas

I have not been to Opatija, but the other ports would be easy to do on your own with a guidebook. We went on a Star Clippers cruise and did a mix of shore excursions and independent touring but could have easily done it all with the guidebook. We used the Rick Steves Croatia & Slovenia guidebook and it covered all of our stops including Kotor as it is considered a daytrip from Dubrovnik.

In Kotor you can also go take a bus to Perast which is another small town with an island that has an interesting church. You will pass by Perast on your way to Kotor while on the boat. We did a tour here because it combined Perast and Kotor and we did not want to bother with arranging transportation to Perast on our own. Kotor was one of our favorite stops on the cruise. Very small town and easy to walk around. If you are there after dark, the town wall gets lit up and you can get a great visual of how it surrounds the town up the hillside.

In Hvar our ship docked outside the "new" town which was really small and had a few interesting stops around town. Then you can hike up to the fortress above town which has a small museum. There was an option to go to the old port town Stari Grad which we did not do.

Dubrovnik is a little bigger and very crowded but still well covered by a guidebook. You can hike the town wall and there is a cable car up to a view point above the town that is great.

Have fun!

Kotor: See old town and perhaps explore some of the wall if you like. Be forewarned! We took a Celebrity Cruise Line excursion from Kotor and encountered the following. There is a certain road that leads away from the bay inland. It goes up the side of a steep mountain and has 25 numbered 180 degree switchback turns and at least 75 other sharp turns. Rock wall on one side and nothing on the other. Can't remember the name or number of the road, but it went to the towns of Njegusi and Cetinje. Don't mean to rain on your parade, but don't want you to get into a hairy situation without any knowledge. If anyone in your party is prone to motion sickness or has a problem with heights you might want to avoid this road. Several people on our tour bus real problems with the heights and switchbacks. You can go to bing.com, use the map feature, and see aerial views of the road from &*%$. In Dubrovnik, just walk the wall, have lunch, a cold beer, and enjoy the people watching.

Don't know where in Opatija the ship docks, but it is doable on your own. Just walk along the water and see the Beautiful hotels, sit on one of the terraces and have a drink.

In Dubrovnik walk the wall, will take about 1-2 hours. Wear a hat because there is very little shade. Walk the streets of Old Town. They have quite a few good museums; they are small but you don't need too much time.

Dubrovnik requires even small ships to use the purpose-built cruise port, but the city bus liberates you from the ship's infrequent shuttles. I would not pay for an excursion in any of these towns, and I have paid for plenty of cruise excursions. If you can possibly walk ashore after dinner, the cities are very different after the daytrippers have left. (This is a good place to pay for dinner ashore, if it works out for you.) I particularly remember people dancing in the dark (in Kotor), the Kotor Sea Organ, and a hotly-contested local basketball game under lights. (I think Basketball is #2 after water polo, in Croatia.) Kotor also has a Maraska (maraschino) liqueur factory, but we didn't bother to visit.

Opatjia is very different, there is little old town, much beach, and the ghosts of the Austro-Hungarian empire, for which this was their Riviera. All of Croatia has exceptionally clean sea swimming.

I haven't been to Hvar, but the internet pictures look the least interesting. I've been to the others, on a 22-cabin ship that moored right in front of most of the city walls (except Dubrovnik.)

Kotor, Dubrovnik: Been to both. Dubrovnik is the large of the two and is survivable when the boats are in town. Still a little Disneyland like. Kotor is very similar to Dubrovnik, but since I got out before the boats unloaded I cant judge what your experience will be like. My guess, not pleasant given the confines of the place. If you are only going to do one, it has to be Dubrovnik. Now if you can get a special trip along the bay from Kotor towards Perlast and further north, then that's worth the effort.

How long are you in port at each of these stops? Regarding the shore excursions offered, I would only consider them if the excursion goes out beyond the confines of the local area (e.g. a Kotor excursion that goes somewhere outside of the town). All of your ports are areas that you can easily explore on your own.