molly,
You are not crazy. We are in our 80's and have almost always driven in France. Our last trip we did take some trains for longer distances because it worked out better time-wise in moving from base to base, but we got rentals from two locations (Marseille and Bordeaux). In planning our routes ahead of time, we always use a Michelin paper map for an area to plot out our routes and where we want to stop. Then I check that route on viamichelin.com to see the road #s, towns and sights along the way, travel times, etc., as well as choosing which route between places. (Some are strictly autoroutes, the A roads...faster but often more boring and expensive with tolls. Some show that they are scenic.) We almost always pick the more scenic routes (the D roads).
We have never had trouble finding gas and rest stops, although in a tiny town, with one gas station, you may find limited hours there. On autoroutes there are larger stops with gas, food etc., just as in the US. As others have suggested, pay attention to the road sign directions if there is a conflict with your online road instructions. Know the rules for roundabouts and learn the road signs (not difficult at all), pay attention to speed limits, know ahead of time where you will park the car in whatever town or city you stay in. Best if you have lodging that includes parking.
The drive from the Arles area to Sarlat may be long, but plan on a lunch stop for an hour or so to break up the drive. Having a car in Provence is much better for exploring and experiencing all there is to see. It is really essential in the Dordogne and Lot valleys. When we were 79 and 80 we spent 10 nights in a gite near La Roque Gageac in the Dordogne Valley and drove all over to see what we had planned to see and do. We didn't have any drives there that were more than an hour. We had no problem there. (A caveat...Sarlat can be a pain to get into and out of. We chose not to stay there because it is more tourist-heavy and trafficky. Our gite was a 15 minute drive away. We only went to the town of Sarlat 3 times...to visit the market on a Wed., to have a fancy dinner one night, and to have a simple dinner our last night in the area and find an ATM for some cash.)
I would keep the car and drive to the Loire. Find someplace to stop for lunch or sightseeing along the way (e.g. Oradour-sur-Glane). Count this a part of your sightseeing as well. Look at that Michelin map for the overall view.
How long is your trip? I ask because 3 nights in a place means only 2 full days there. Could you extend any of these stops? Provence, not including the Riviera is over 10,000 square miles, of which the Luberon is about 230 square miles. I would pick the things I want to focus on to plot out any driving routes. Likewise in the Dordogne. Two full days there is really, really short. However, in two full days you could get a taste of the area (The Dordogne region is almost 3500 square miles, but some popular things to visit aren't too far apart by car. Choose wisely to not feel rushed or frustrated.
With two days in the Loire you can comfortably visit 2 chateux in a day and still have a relaxed time. For me, I wouldn't want to see more than three chateaux because I would have chateau overload. There are other ways to spend time there (wineries, mushroom caves, troglodyte sites, gardens...) Bayeux is a good base for the DDay sights. Two full days give you a full day tour or drive to the beaches, with a stop at one of the museums in the area. If possible, be sure to be at the American cemetery for the flag-lowering ceremony...very moving. The museum in Bayeux and the church there are good. Alas the tapestry is not there at the moment. A drive to visit some cider-making operations can be fun. Check the forum for ideas about Normandy (11,000 sq. mi, includes Rouen and the drive along the Seine, not just WWII sights.)
Hope this helps!
