Annecy

Besancon is not a very interesting place, but if you are interested in WW2 it has a fortified castle which was Gestapo HQ. Really interestingly historically, but not for younger children perhaps.
Mulhouse has the national rail museum and a classic car museum, both worth a visit.

If you are going to do stuff around the lake, and not just walk around the old town,
I might consider extending Annecy. On the other hand, if all you want to do is stroll
the town and down to the lake, 3 nights might be overkill.

Lausanne has the Olympic Museum, and is good enough to kill a day. If you took
a day off of Annecy, you could go a bit out of the way and hit Lucerne.

If you wanted a day or two in the Alps, the Portes du Soleil area is an hour or so from
Annecy. Morzine is a nice town to spend a night or two in.

Is there a reason you are starting in Zurich if all of your first stops are in France? Could you fly into Geneva?
I thought 3 nights in Annecy was perfect. We biked the lake and explored one day, and drove to Chamonix one day. That, plus a travel day was perfect.
Are you seeing anything else in Switzerland? If you can squeeze one more day, 2 nights in Lausanne would be great. The Olympic museum was fascinating, you can take the paddle wheel boat (or drive, but the boat is more fun!) to Montreux and the Chateau de Chillon or even take the boat across to Evian les Bains. If you have more time - a trip up to Gruyère or the chocolate factory . There is so much to see that is easily accessible from Lausanne. That would be my vote!

Is there a reason you are starting in Zurich if all of your first stops are in France? Could you fly into Geneva?

Depending on where you live in the US, Zurich many times has more direct flights than Geneva. I am guessing because of the car drop in Basel, the OP may be doing a river cruise after visiting France, so landing in Zurich is not an issue. Just guessing.

Joanne, I would suggest you do your trip in reverse if you can. Also, unless you need a car, all these places can be visited by train very easily and save you the cost of a car rental and all the associated hassles and costs.

Zurich
Lausanne (2 nights)
Annecy (2 nights)
Lyon (2 nights)
Dijon (2 nights)
Basel (1 night)

I enjoyed spending a day in Arbois. There’s an excellent hike (search up tuf waterfall arbois hike) and an excellent Louis Pasteur museum/house.

Chamonix and Mont Blanc are so close. Well worth a day trip.
Cathy