Dordogne + French Riviera?

You did not specify how many days/nights you would have for the entire trip.

We have done that spending a week in a village in the Dordogne with friends then driving to Roquebrune Cap Martin where we stayed in the ancient village for a couple of weeks in a friends place. If you want to explore the Riviera Nice would be an excellent base and the train runs along the coast so you would not need to keep the car and could train back to Paris. We spent the night in Arles on our trip to Roquebrune. Driving and parking are difficult in the Riviera. You do need a car in the Dordogne and we prefer a village as. a bade. We have stayed in Limeuil and in Cadouin and then driven to gardens, chateaux, villages and kayak launch sites from there.

Marlene,
How long is your entire trip? You will definitely be better off with at least 7 nights in the Dordogne. And with a car, of course. Factor in a day to move from the Loire to the Dordogne. Yes, it is listed as about 4 1/2 hours drive (e.g. Amboise to Sarla-le-Caneda), but that doesn't include stops for food/gas/comfort/tolls. It all adds up, plus there is checkout and check in times, etc. Also figure a day to move to the Riviera from the Dordogne, whether by car or train. Add in about 5 days based in, say, Nice. Allowing enough time for moving to a new area of France makes that time less stressful, whether you drive or take the train.
So, to repeat, how many days/nights do you have in France?

As part of a trip to France in Aug 2021 we stayed first in the Loire Valley (3 nights), then rented a car and drove to Dordogne (6-7 nights). We then visited other areas. Our trip report might be useful reference: https://community.ricksteves.com/travel-forum/trip-reports/trip-report-france-28-days-aug-sep-loire-dordogne-languedoc-roussillon-lyon-burgundy-honfleur

Marlene.
If you forgo the Riviera, the return from the Dordogne to Paris is easy the way you have considered. We drove from the Dordogne area (near La Roque Gageac) to Bordeaux where we returned the car at the train station, spent two nights, using the one full day to see some sights in Bordeaux, then took the TGV to Paris (just over two hours trip). We spent our last night at a hotel at CDG since our flight the next day was a bit early.

We did a Provence / Dordogne / Paris trip a few years ago. No reason you couldn't do the same , subbing in the Riviera instead. Fly Paris to an airport in the Riviera area. Rent a car, explore. Then drive to the Dordogne. For that bit of the trip, we drove to Carcassonne for two nights (explored Cathar castles on the full day there, enjoyed Carcassonne in the evening and early next morning), then headed to Sarlat for four nights. We returned our rental car in Brive, then took a slower train to Paris. You could also return your car in Bordeaux if you are OK navigating larger cities. Rental car fees for returning in a different location, but in the same country, are somewhere between $50 and $100, not huge despite what others may tell you.

Sarlat is an excellent base, despite being crowded mid-day. Lots of medieval charm and many restaurant and accommodation choices. Just head out on day trips to avoid the mid-day crowds and do NOT arrive in Sarlat on a Saturday - the market crowds make parking and navigating closed streets a PITA.

Sorry I left out such an important piece of info! We'll have two weeks for this trip total. Thanks!

Marlene,
With two weeks (14 days/nights), I would probably drop either the Loire or the Riviera. I would go immediately to the Dordogne by way of Bordeaux. (There is a tgv from CDG to Bordeaux,but I don't know the schedule.) Or buy your tickets from the US to Bordeaux, changing planes at CDG. Our daughter did this in 2023. Stay the first night in Bordeaux to rejuvenate and drive to the Dordogne on day 2 (about 3-3 1/2 hours). Spend nights 2 through 8 there (6 full days). Return to Bordeaux on day 9, fly to Nice from there (not expensive). Car rentals are available at the train station and the airport in Bordeaux, so easy pick-up and return for you. Stay nights 9 through 12 in Nice (3 full days), or other Riviera town. No car needed here. That gives you all days 10, 11, and 12 on the Riviera. Take the train on day 13 to Paris (sleep there) to be near to the airport for flight home on day 14.
This means no Loire Valley or much time in Paris. It just seems all three places are too distant from each other to see/do much. The alternative is to swap out the Riviera for the Loire Valley. Either way, I would opt for the Dordogne first. And 3 nights is really shortchanging the Dordogne. It isn't as compact as the Loire or even the Riviera. It deserves a minimum of five full days (10 is even better!) I have been to all three areas (some more than once) and would put the Dordogne first and for longer time.
Perhaps you could make a list of your priorities of things to see and do in each of these areas to allocate the time needed in them. Remember, the Dordogne isn't compact like the Riviera, or even the Loire Valley, so you do need more time there. The lists may help you decide on your itinerary. Best of luck!

Just to clarify, our trip will be either:
Paris-Loire-Dordogne
OR
Provence/Riviera-Dordogne
Thanks!

All these places are interesting -- and it is a matter of personal taste. I am not a big fan of Provence compared to the others. I'd probably do the 3 nights in the Loire for the castles, then drive to the Dordogne keeping the car and base in a smaller town (so much easier to park and drive in and out when you want to spend your days driving to caves, to chateaux and gardens etc and the charming villages). I'd rent a place for a week in the Dordogne, then spend your last 4 nights in Nice flying back from there.

We have flown open jaw for decades and it saves money and time. You book on 'Multi city' on the airline sites. Just in the last couple years we have flown into Milan and out of Paris, into Paris and out of Limerick ireland and into Paris and out of Berlin.