Car rental companies do not usually offer cars with toll transponders (Badge), moreover the additional cost of this option would only be of interest if you constantly drive on motorways. Lanes dedicated only to electronic tolls (Telepeage) are located at the left or right ends of the toll and have a "T" sign.
To my knowledge, there are no more manned booths.
Generally you enter toll highways via a toll booth where you are issued a ticket.
When you leave the toll section you insert this ticket into the machine and pay the requested amount.
Lanes allow you to pay either in cash or by credit card (with contactless function), and often have both options. It is indicated by signs representing a credit card or/and bank note and coins.
Just like Judy, between Bordeaux and the Dordogne I advise you not to take the motorway (Autoroute A89) but rather the D936 departmental road wherever you go in the Dordogne. It's not much longer, it's more pleasant, and it will get you used to the small roads in Dordogne where you will no longer find any other highway but rather roads like this:
http://tinyurl.com/ybaa68dw
From the Dordogne to Lyon you can mainly drive on "autoroutes" via the Auvergne region in the center of France (Clermont Ferrand)
I would like to point out that the Sarlat-la-Canéda and La Roque-Gageac area is only a small part located in the extreme southwest of the Dordogne region which is in fact a valley. Get yourself a good travel guide and you will find that you can go much further east to visit this region.
Or check out this map and see where Sarlat and La Roque-Gageac are located:
https://www.vallee-dordogne.com/app/uploads/ot-dordogne/2022/03/carte-vallee-dordorgne.pdf