In March, you'll only see 95% locals, which is great for getting into the Dordogne atmosphere. There are no school holidays in the region at this time of year. Sarlat, which is like Disneyland in the summer, will be a fairly "normal" town with its usual daily life.
There is very little or no agricultural activity before spring. Winter activities like the "marches au gras" (duck and goose product sales) are over, but farmers' markets operate year-round as usual in all the villages.
Apart from a few caves and castles that are open all year, most tourist activities don't start until April. Obviously, there's no canoeing or kayaking on the Dordogne in March.
The weather isn't predictable, but March and sometimes April are rather rainy months (though less rainy than in Finistère, where I advise you to bring waterproof clothing). It can be cold in the morning with some frost. If the weather is sunny, don't expect temperatures above 15°C during the day, but climate change could alter things, who knows?
From what I know about the Bruno Chief of Police novels, don't expect to see the Dordogne region exactly as it appears in those stories.
Book accommodation with heating (or a fireplace!), and,of course don't go to the Dordogne without a car.