The train from Avignon to Nice will take about 4 hours. That gives you 2-1/2 days in Nice, which is plenty for Nice itself unless you feel the need to go to every one of its many art museums. However, most people go to Nice as much to see some of the surrounding coastal towns and hill villages as to see Nice itself. If you want to do much in that regard, 3 nights is just not going to be adequate.
I'd say the same thing about 2 nights Provence. I think you're giving the area short shrift with your plans.
Sometimes it's better to spend more time in one place. It takes more time than one might imagine to change hotels. It's not just the time spent on a train or a bus; you must pack up, check out of your hotel, get to the train or bus station, and then you must repeat that process at the destination.
Where will you be before Paris? After southern France?