Marilyn - We just got back from Croatia a week ago and loved it all. In my opinion, unless you have a lot of day trips planned, a week is way too a long time to stay in Dubrovnik. The town itself is beautiful, but it gets hit most every day with people from all the cruise ships. Two days in town is sufficient and add extra days for day trips.
Plitvice National Park is a do not miss site in my opinion. Stay in the park if you can - we stayed at Hotel Plitvice and it was very convenient. Two things we would have added to our itinerary (but we didn't have time) was to go to Mostar (overnight stop) and a day trip to Montenegro from Dubrovnik. Also, the drive down the Dalmatian coast was beautiful and an easy drive.
It's easy to rent a car and not too expensive as long as you rent and return in Croatia. I've had really good luck with SIXT in several countries including Croatia for one and two day rentals. The cars have always been in excellent condition and they have always charged exactly as quoted - can't say that about other companies. I've also used Kemwel (usually minimum 3 day rental) and Carjet with no issues, but not in Croatia.
I agree with the other posts, you can skip staying in Split. If you really want to see the palace, just head over there for a morning.