Chamonix or Annecy

I'd do an extra night in Chamonix especially if I could do the Heilbronner cable car; if it took two extra nights to do that, do two nights there. And two nights in Annecy (one would also be fine for Annecy -- we enjoyed several days just hanging out, but there isn't much to do -- it is a lovely place and one night is plenty to see and stroll and take a boat on the lake.)

Chamonix is a cute town. I spent two full days there (3 nights). There isn't too much to see. Taking the trip up to see Mt. Blanc and over to Italy was something to do once. I'm glad I did it once, but the price is too much to make it a regular thing. You really need to stay there to get up the mountain first thing in the morning.

Annecy is also a cute town. The old center was nice. I only spent a half day there - drove in early and left after lunch. There are a couple things to do there that I might have enjoyed if I had more time.

I had a rental car from Lyon because I didn't feel the train schedule worked very well for visiting Annecy, then moving on to Chamonix.

I'd add the factor of getting from Chamonix to Annecy (I think the tour
ends in Chamonix?)

Under the best of circumstances, it's pretty easy. You can take Flixbus
or BlaBla bus, which will be comfortable motor coaches and the ride is
60-90 minutes. The train will get you there also, but there are a couple
of connections and it takes a bit longer. But the bus schedules are not
always the same each day, so you will want to check pretty soon.

If you'll already be in Chamonix, then I'd say spend one more full day &
night, transit to Annecy the next morning, and you'll have a day and a
half in Annecy. Flixbus & BlaBla bus are probably the easiest ways to get
to GVA from Annecy after that, and there is also the 272 local bus.

We will need to get from Annecy to Chamonix on October 19th. We are two seniors in our late 70's who do not want to hassle with 3 or 4 changes if we book trains for this part of our journey. I was thinking about trying to find a private car/taxi to take us, but wonder how difficult it would be to take the bus as you have suggested? Worried that the steps are high so may be hard to take our luggage on board? We will have two carry-ons, a backpack and purse. Can we book seats in advance so we can sit together? Thanks for your advice!

I took this RS tour (fabulous!) in early June. At the orientation walk in Chamonix the guide showed those interested where the bus stop was. It was about a 10 minute level walk from the tour hotel, basically one end of the long main street to the other.
I reserved online for the BlaBla Bus to Annency. I did reserve a seat for a small additional fee but the driver seemed at first to not want to honor it? Showing the ticket seemed to make a difference.
The luggage is loaded underneath, like the RS bus. so you wouldn't have to get yours up the stairs of the coach, which was an easy, clean and comfortable ride. I spent that afternoon and evening in Chamonix, and it would have been fine to depart in the afternoon, if you opt for that, but I had an extra day.
The train and bus station is at the same location, for your Geneva leg.

Note, the bus station in downtown Geneva is not colocated with Cornavin
train station. It's about 4 blocks away towards the lake, so not too far, but far
enough to be an issue if you are running late to catch a train.

I believe the local buses and trams do stop at the train station, but buses like
Flix, BlaBla, and Swisstours drop off at Pl Dorciere, and those are the buses
you would most likely take back from Annecy.

Now, if you're talking about Geneva Airport, then, yes, the airport, train station,
and bus station are all in the same place, more or less.

As noted, the bus stop (there isn't really a station) in Chamonix is at the intersection
of Av de Courmayeur and Rte des Pelerins, just a bit west of the Aiguille du Midi
plaza.