I don't know the lifestyle to which you are accustomed, but I don't think you should find it hard to stay within your price range. It depends, of course, on what you're looking for.
Take a look at La Villa de Consuls in Sarlat-le-Caneda for your stay in the Dordogne. Near caves and I think you can rent a small apartment within your price range. If you can be in Sarlat on a Wednesday or a Saturday, the Old Town becomes a marketplace, with performers scattered throughout. My grand daughters (9 and 12 at the time) were big fans.
A much plainer place which our grand daughters enjoyed was La Chevalerie, outside of Amboise (and therefore near several chateaux). It has a swing set, duck pond and a donkey named Pamplemousse (Grapefruit), who absolutely captivated our girls. We still think he was hee-hawing "goodbye" when we drove away. Little English spoken, but we had no real problems communicating with gestures and a few words of each others' languages.
Both of these are listed by RS in his book on France.
You might want to consider visiting Chaumont-sur-Loire. They have a garden festival which includes lots of spaces done up by different landscape artists. Kids won't like them all but will find some fascinating (adults likewise). When we were there, they had a small ravine that was rigged to fill up with mist every few minutes. Our girls could have stayed there all day. I can't say they'll repeat that, but look for it just in case.