Thanks, Sandancisco, for mentioning my report. While the Dordogne was amazing, I think it's relatively difficult to get around without a car., although some forum members have done so. Others on the forum have suggested Avignon or Arles in Provence for people traveling without a car, with access to places like Aix-en-Provence, St Rémy, Ile sur la Sorgue, Nîmes, and the Pont du Gard. Or you could head north, where it might be cooler, to Lyon and Annecy or Chamonix and then Alsace. I'm headed to Alsace for 5 nights this week, with no need for a car. I could easily find enough to do for a longer stay, including day trips into Germany if that would be of interest,
There are a number of forum members with extensive experience traveling in France without a car. I'm sure some of them will weigh in.
Edited to add that a search on the forum for France without a car gives some useful results. It would help to know if you are hoping to go to just one or two bases for the 2 weeks, or if you are interested in shorter stays.