Suggestions for Four nights in France

I would say train to Avignon. Easy to pick up rental car at the TGV station upon arrival. You could stay Avignon or another town like Arles or Aix-en-Provence. Around there are the great Roman sites like Glanum and Pont du Gard, the Van Gogh sites, Cezanne sites, castle ruins like Les Baux. We spent a month in the area two years ago and did something different every day. And the weather will likely be more mild than Normandy. Use wunderground.com weather "history" feature to see what past years have been like at that time.

I'm not at all sure that you will like the Dordogne in December/January, it's a region that can be visited when the weather is nice. In winter you might find it boring, even depressing and I wouldn't want you to be disappointed.

https://youtu.be/Zd9ZKV1gYyY

Many places to see are closed, Sarlat which is like a "Dordogne-Land" full of tourists in summer is totally quiet in winter, the only traditional local activity which attracts people is the "marchés au gras" (foie gras market) .

The weather is cold, and you might even get snow, which, given the road infrastructure in the Dordogne will make getting around more difficult.

Normandy will probably be just as depressing and probably colder, windy and rainy.

Instead try the regions on the Mediterranean coast, Provence, Cote d'Azur, but it will take 7 to 8 hours to drive there from Paris

What you have heard about the Normandie region is pretty true sunrise late, sunset early and messy weather for days at a time. I love it there but winter is no fun.

Sarlat sans tourists ought to be pretty nice. Thumbs up for your decisions.

I agree with suggestions of Avignon. As others have said, it’s an easy train ride from Paris and you can pick up/drop off rental car at TGV station. Sarlat and greater Dordogne region tends to be cold and rainy in winter. Also most activities are limited, if not closed.

Although the Dordogne is my favorite area in France, I would hesitate to spend time there in the heart of winter. Also, there is no TGV so it takes a while to get from Paris to the Dordogne.
I can’t remember, but if there is a TGV to Nice I would consider Nice . There is a reason, British and Russian royalty wintered there.

Chartes is a good visit for at least one night. A short loop through the Loir Valley is probably the more convenient visit for four days with a rental car. I'd suggest staying in Amboise and visiting a few castles in the area.

Bear in mind that 1 January is a public holiday. 31 December is a big celebration night - need to book restaurant.
Think about tgv to Bordeaux- beautiful city, walkable, great art gallery and wine museum,