We were in Florence mid-may last year. My wife and son wanted to climb the dome, and we chose not to try to reserve spots before leaving home. Our RS tour guide told everyone in Venice not to obsess about reserving until we got to Florence. Once there, we went to the ticket office and bought Duomo tickets and reserved times for the climb, using the self-service kiosks. Spots were available for the following afternoon, when we knew we had free time. It was an easy process and my wife and son really enjoyed the climb and the view.
We had been checking fairly regularly before leaving home and it seemed there were usually spots available for the following several days, though they often were all booked for the same day we looked. I think many, many, many people don't book until they're in Florence, so as long as you have a little flexibility, I'd imagine you could find available times during your three days in the city. They move a lot of people through there every day. And make sure you hang onto both the Duomo combo ticket AND the slip with your reservation time printed on it. Otherwise, you'll be turned away.
BTW, we totally fell in love with Florence from the moment we arrived. Have a great time!