Hello! Yes, Dubrovnik in June will definitely be crowded (and quite expensive), but it surely is a beautiful city worth seeing. I don't think you'll regret going there, but it's up to you to decide if it would be worth enduring the crowd. :)
As for the other towns, I definitely suggest that you visit Split and its near islands. It's vivid and perfect for nightlife and cultural enrichment. If you're less into nightlife and more into spending time exploring nature, you should come to my island, Vis. It's replete with beautiful untouched secluded beaches and it's less crowdy and commercialized than other islands. I'm sure you'll have a great time whatever you decide. Have a nice day! :)