We fly into Dubrovnik 25 June.
We fly out of Split 2 July.
I would like to see Dubrovnik, Korkula, Hvar.
I also would like to visit Mostar--hoped to spend the night there.
Is there enough time to do all of these things?
Or should we give something up?

It depends on what you want to experience.
In my opinion it is very difficult to experience any of these places just by staying there a day and taking a walk along the main turistic street. For the islands, I would recommend you to take at least two-three days and also go inland to see places less exposed to changes due to hotels, restorants etc. Split you can go through in a day, it is enough for a quick tour through the old city.
Evenings in Dubrovnik are lovely, enjoy few evenings there instead of running around, especially if you are young couple, I got very nice memories from the place.

To cover that amount of locations you are going to spend most of your time travelling and worrying about ferries and places to stay.  If you are happy being on the move then fine, but it will not be relaxing.

Well, we are not a young couple. 
We are early 50s, seasoned travelers, and will have our 16 year old son with us.
In general, we do like to be somewhat on the move, but not *frantically* going from place to place each night.
I had planned three nights in Dubrovnik, but had heard really good things about Mostar. Perhaps we could do a day trip.