First thing I’d say if that I don’t see the point in saying “this part of France is also nice you should try it” as basically the whole of France is nice (and yeah I know I’m a homer but it’s still true).
Second, there are two ways to travel: either you do the whole thing by car (except maybe the very last leg of the trip) and in this case it becomes a road trip, much more flexible but with its own constraints of having to pack/unpack pretty much on a daily basis; or you select 2 or 3 “bases” that you can reach by train or plane and then explore from there by renting a car for a few days.
Third, and I’m going to simplify a bit, but you have to understand there’s no transportation going East-West through the center of France. You have to imagine a loop that’d look a bit like a raindrop, and that goes Paris → Lyon → Avignon → Montpellier → Toulouse → Bordeaux → Paris. That’s the way the transportation network is designed. So basically there’s no direct way to go from Bourgogne to Dordogne. It’s either Bourgogne → Paris → Bordeaux/Toulouse → Dordogne, or Burgundy → Lyon → Montpellier → Toulouse → Dordogne. So I’d recommend planning around this loop either clockwise or counter-clockwise.
Taking this into account I could imagine 3 possibilities, knowing that they all end with Paris for a week.
Itinerary 1: the Southwest
Tours (3) [Loire valley castles, Chinon…]
Bordeaux (4) [this city is much nicer than it used to be]
San Sebastian (3) [2.5 hours by car from Bordeaux]
Dordogne (4) [2+ hours by car from Bordeaux]
[direct train back Bordeaux-Paris ~3 hours]
Itinerary 2: the South-East
Burgundy (4)
Lyon (2)
Avignon/Provence (4)
Côte d’Azur (4) [fly back to Paris from Nice, or direct train back from Marseille, reach Paris in 3.5 hours]
Re: this itinerary there’s a lot of flexbility. You can go directly from Beaune to Avignon (2 hours and 40 minutes if choosing the right train) or stop in Lyon. Youcan limit yourself to the area “around” Avignon (Orange, Pont du Gard [must see], St Remy de Provence, Les Baux de Provence, Aix-en-Provence Luberon [especially Roussillon], Ventoux, Camargue…) and move progressively towards Marseille / Toulon (Cassis, Porquerolles, Bandol) and go back to Paris by train. Or you go can farther to Cannes, Antibes, Nice, Monaco for a different experience (and no wine). And if you want a bit more variety and figure that 8-10 days in the South of France is more than you want, you can always add Champagne before Burgundy…
Itinerary 3: a bit of both
Burgundy (4)
Avignon/Provence (3)
Toulouse (2) [Avignon → Toulouse = 3.5 hours by train direct]
Dordogne (3) [Toulouse → Dordogne = 2 hours by car]
Bordeaux (2) [train back to Paris ~3 hours direct]
Re: itinerary #3, Avignon → Bordeaux is a bit long in one go (5.5 hours by train, roughly) so I think it’s best to take a pause somewhere, either Toulouse as suggested or in the Languedoc (Montpellier to enjoy the sea, Carcassonne to visit the incredible medieval town…) and then push on. You can also take a direct train back from Toulouse to Paris (5.5 hours) or fly back.