Favorite places near Nice and Avignon

I'm assuming you'll have a car since you're asking about places less visited.

This is a second-hand suggestion because I (without a car) didn't manage to get there: The village of Tourrettes-sur-Loup (north of Vence) was recommended to me by a local I happened to chat with in Vence. She said Tourrettes is lovely and gets very few tourists.

Vence itself has a rather large historic area (a bit of a walk from the bus stop) where I don't think I saw another tourist. Actually, there were not many people out on the street at all, so if you seek liveliness, the afternoon siesta period should perhaps be avoided. It didn't bother me to be the only one walking around since I had arrived from St-Paul-de-Vence, which was exactly the opposite. (Very few of the people in St-Paul-de-Vence bothered to go to the Fondation Maeght, so you need have no hesitancy about seeing it if you're art lovers.

I went to Biot, another hill village, because I'm interested in glass and it has a small glass museum. There's also the Leger Museum on the edge of town. Biot is pleasant enough but not spectacular. I'd say the same for Vallauris, which is known for pottery. It's where Picasso did at least some of his ceramic work. Both of those places have a few shops obviously targeting tourists, but not unpleasantly so.

On an extremely rainy day I took the Train des Pignes (not exactly cheap) to the fortified village of Entrevaux. I think it must surely be on the tourist circuit for those equipped with cars; it certainly has the look. Since it was raining cats and dogs, I was the only visitor there, so I can't say how touristy it might be under more normal conditions.

Not necessarily "less visited", but from Menton it makes sense to visit places across the Italian border too.
The Val Nervia is especially beautiful; the main highlight is Dolceacqua but it is worth going deeper into the mountains, to Apricale at least.
Sanremo, a sizeable coastal town, is also very charming.

I too recommend Dolceacqua and Apricale, Vence (& St. Paul de Vence) and Tourettes Sur Loup (Continue on to Gourdon). You'll probably go to the Friday market in Ventimiglia but I also like the seaside town of Bordighera. The old town above Finale Ligure is also nice, as is the drive down the coast.

You'll likely go to Eze but continue past to La Turbie (and for the more adventurous, to the Tete du Chien --- a hike). Also on the loop is Peillon, a lovely stone village with a couple of nice restaurants (big sister Peille isn't quite as charming). Villefrance-sur-Mer is quite nice (but hugely popular) as is Cap Ferrat (Ville Ephrussi or simply the walk around the Cap).

You can also head to the hill towns like Breil-sur-Roya, Saorge and Tende (note road closures or take the train). This part of France was the most recent border shift (in 1947) and the Italian influence is as strong as the espresso. .

If you are depending on public transit, things are still possible, but connections will generally be via Nice (for things like the train des pignes and the train des merveilles).

What a wealth of information - and especially of the smaller less touristy places the three of you mentioned. i will share my experiences when i return in August.