I just came back from Croatia and in Dubrovnik, I stayed at a sobe recommended in Rick's Slovenia and Croatia book, Apartments Pavisa. The owner, Pero Pavisa, was great and there was a little kitchenette in my room (i think all of his rooms). It's right by Buza Gate so if you stay there, make sure you get dropped off at that gate rather than Pile Gate so you don't have to haul your stuff up a zillion stairs. It's at the top of town, so the climb is not fun, but I liked it.
in Split, I was supposed to stay at Kaleta Apartments in Diocletian's palace, but I think she double booked me and subsequently gave my room away. It was very strange and she gave me a song and dance which was pretty sketchy, so I don't recommend them even though they are also in Rick's book. I ended up staying in Golden Gate Rooms, and had a nice room with a little balcony right in one of the walls of the palace by Golden Gate. No kitchenette, but I didn't need it anyway.
Enjoy your trip!