I rented a car on arrival day and departure day only....and would do it again. It was just me and my 9 year old niece, so we only needed a compact. Rented from Dollar, I don't remember the exact price, but it was something like $23 total for each rental. I went to their website and signed up ahead of time for Fastlane....when I picked up the car at the airport, there was no line for Fastlane members, it took a total of 5-10 min at the counter, then walked across and got in the car and drove away. We stopped at the grocery store, then drove to the resort, checked in, drove to the parking lot near our room and unloaded. Later that day we drove to the oultlet mall later (we would not have gone without a car). Dollar has a counter at the Downtown Disney Courtyard by Marriott, so we drove there in the early evening and dropped it off (they close the counter at 4 or 5pm, but you can drop it off and leave your keys and paperwork in the drop box. I only kept the car a few hours, but could have had it for 24 hours, but we were going to the parks bright and early the next day. We walked from the Marriott to DD and since we were staying at POR, we took the boat back (we wanted to ride it anyway).....we could also have taken the Disney bus. On the day we left, we went back to the Marriott and picked up the car, loaded the luggage and checked out, then spent the rest of our time at the pool until we had to drive to the airport.
You'll need a bigger vehicle, so you'll pay more, but check for the discounts and you may still come out cheaper than a towncar (price+tip). Since you have 2 adults, you'll have more flexibility that we did. We stopped for groceries on the way, but with two adults, instead of dragging the whole family to the grocery store (if you planned to stop at the grocery store) you could go straight to the resort and check in, then have one of you stay with the kids at the pool or explore around the resort while the other goes grocery shopping. When you return your car, you can all go if you want to spend some time at DD or just one of you can return the car while the other stays back at the hotel with the kids.