where to stay in Dordogne region

2.5 days for the Dordogne is a very short time. You'll have to be smart and make choices and concessions.

Sarlat and its immediate surroundings are only a tiny part of the Dordogne (and the most crowded with tourists, go figure...).

To give you an idea, take a look at this map:

https://www.google.com/maps/d/edit?mid=1OdPOg8AgsNn0Jlv1cNHkujaWv9p_Jpc&usp=sharing

If you're smart and have a car, you'll avoid staying in Sarlat, where it's hell to get around and park in summer.

See Sarlat's website about parking in town (click on the British flag for the English version):

https://www.sarlat.fr/mon-quotidien/transports-et-stationnement/stationner-a-sarlat/

The first concession is to skip Rocamadour.
Like most visitors, you're going to be very disappointed by this tourists circus comming by coaches and organized groups from all over Europe, strolling through the narrow streets of Rocamadour with their “made in China” souvenir stores.

If you've looked at the map I mentioned above, you'll see that there are better things to do and see than going to Rocamadour.

If you take the A20 freeway into the Dordogne, there's a village where you can stop for lunch: Vatan, just under 3 hours from Paris.
You'll see signs on the freeway saying “Vatan Village Etape”.

https://maps.app.goo.gl/1rRfX5uXoz8jmqXS7

But the smartest option would be to stop at one of the highway service stations, they all have food and beverage services.

Any other sightseeing suggestions?

See the map.

We spent a week in the Dordogne last Fall and loved it. I'll agree with the above that Rocamadour is overrated. We stayed in Beynac with a view of the castle above. We had a good day visiting the Beynac Castle and then a short drive later Castelnaud which you can see across the river.

We didn't visit Lascaux this trip but have seen it before and loved it. Here's my Trip Report if you're interested. https://community.ricksteves.com/travel-forum/trip-reports/trip-report-part-2-hills-and-castles-dordogne

Betsy,
By 2.5 days I am thinking that you are arriving about mid-day and staying 3 nights (2 full days) then leaving. Is that correct? I totally agree with skipping Rocamadour. And staying in Sarlat means some difficulty and time-wasting getting in and out of the town. Where to stay.....there is Beynac-et-Cazenac, La Roque Gageac, Les Eyzies, Domme, Limeuil, St, Cyprien, Belves, and many others. In one day you can visit a castle in the morning and a prehistoric cave in the afternoon, or vice-versa. The 2 best castles are Castelnaud-la-Chapelle and Beynac. Each will need a minimum of 2 hours. The caves will vary, time-wise. Lascaux IV is a wonderful reproduction and has an excellent exhibition area with tons of information and displays. We spent a total of 4 hours there (there were 7 of us, ages 9 to 80). I don't turn my nose up at reproductions because the presence of visitors in the past did much to harm and destroy the prehistoric artwork in the real caves. However, there are caves you can visit that are original...Peche Merle, Font de Gaume, Chauvet...I don't remember them all off the top of my head. Reserving a time is a must for a visit, but I can't imagine going to the Dordogne without visiting at least one cave, original or reproduction.
A good kayaking company we used was Canoes Loisires in Vitrac. Plan on 3-4 hours for the peaceful and beautiful trip on the river. Make a reservation. We went in the afternoon. You could visit a market in one of the towns in the morning (go early), even the big one in Sarlat (on a Wednesday or Saturday), get something to eat for lunch, from a market stall perhaps, and kayak in the afternoon. In Les Eyzies there is a wonderful museum of Prehistory, and the Abri Cro-Magnon.
You can seek out agricultural places to visit (farms, orchards, vineyards). Personally I wouldn't hesitate to visit a goose farm that produced foie gras, but I realize that is not for everyone. Food products from the area are honey, truffles, walnuts, duck, strawberries, mushrooms, lamb, plus Rocamadour cheese (among others). Monbazillac wines are local, but a little sweet for my taste. There are other wines produced in the area also.
You will easily fill your 2.5 days. I haven't mentioned any of the gardens and chateaux in the area, or things along the Vizere river. When we went, we rented a large house for 10 days through Gites-de-France, so I can't help with hotel suggestions. One place we had a nice lunch at was in La Roque Gageac, called Les Courtines. We also had a splurge dinner at Le Grand Bleu in Sarlat. We ate at our house most breakfasts and some dinners. The cafe at Lascaux IV was good, and one by Beynac Castle also. Other meals were at small local cafes, whose names I don't recall.
The region has a large amount of "the most beautiful villages in France".
Oh, yes....Our rental was part of a large chateaux complex, completely up-to-date...4 bedrooms, 3 baths, parking, pool, modern, fully outfitted kitchen, washer and dryer, easy walk down to the river, orchard on the property, caretakers on the property....$3000.00 for 10 nights for 7 people. 100 euros for the caretakers to clean up after we left. You may find something in your budget, but the French people travel to the area themselves in the summer, and lots of places are reserved well in advance.

We all loved our stay there and would go back in a heartbeat! Bonne chance!

P.S. If you like pate, this is the best place to get some!

Place to stay: Hotel Restaurant du Chateau in Beynac (https://www.hotelduchateau.fr/en). Across the road from the river, at the base of the hill leading to the chateau. Easy access to anywhere in the region. Nice restaurant. Pool.

Sightseeing: In addition to all the stuff you've probably heard of, my favourite off-piste sight was La Roque Saint-Christophe. Medieval village set in a niche of a cliff. Between Lascaux/Montignac and Les Ezyies/Prehistoric Museum/Font de Gaume. https://www.roque-st-christophe.com/

If you follow darrenblois’ site suggestions of visiting Les Eyzies (with both Font du Gaume and a wonderful little museum with prehistoric objects, tools and skeletons of extinct animals), Lascaux and the St. Christophe cliff dwellings between Lascaux and Les Eyzies, then consider staying at La Rosarie in Montignac. https://www.laroseraie-hotel.com/en. All these sites that darrenblois mentioned are along the Vezere River.

La Rosarie us a charming hotel with lovely gardens and a decent size swimming pool. That would give your Dordogne excursion a nice solid root in the Cro Magnon history and caves and cave paintings along with a swimming break or two at the hotel.

Your day trip excursion could go the other way - to Beynac, Domme and Roque Gageac. Kayak on the Dordogne and perhaps Sarlat in the middle (between the Vezere and Dordogne rivers)

If you want to focus more on the castle at Beynac and kayaking on the Dordogne River, Darrenblois’ suggestion to stay in Beynac is sound. What does your family prefer? There is probably a good, active 3 to 5 days of activity for your 2.5 days.

You might want to look into staying at the Hôtel l'Esplanade in Domme. It's a good location and is convenient to several sights. I wouldn't advise eating there, though. Both breakfast and dinner were underwhelming for us. There are better places in the area for meals.