A lot of folks here have talked about how Dubrovnik is one of the cities being ruined by overtourism. While it was crowded when I was there in May 2014, I didn't think it was so bad that I would recommend skipping it. I actually loved Dubrovnik. I found ways to avoid the worst of the crowds for part of the time at least -- the farther you get from the Stradun (the main street through the middle of the old town) the smaller the crowds. And walking the walls first thing in the morning was an absolute delight. I didn't pass or get passed by a single other person for 3/4 of the walk. I think your kids might get a kick out of some of the things Dubrovnik has to offer, but I also agree with valadelphia that you can skip it if you're happy with the rest of your itinerary. I think we all get too hung up on what we might miss when we travel that we forget to just be in the moment and love where we are.
You can also visit large portions of Croatia without stepping foot on a beach. I'm like you, not much of a beach person, and I loved Croatia, including Hvar and Korcula. There's plenty to see and do without being at the beach. On the other hand, it was not so hot in May as it will be in July when you're there, and cooling off in the water might be a welcome respite.
While a car is very helpful, especially with a family of 4, if you visit the islands you won't need the car, and ferries can get you wherever you want to go, including Dubrovnik.
If you need more specific help, it would be great if you can post your working itinerary.