Best places to see for limited time in France

I love all those places. The Dordogne is probably my favorite of those you list but Normandy and Burgundy are a close second. But since you are enamored of chateaux, the Loire is a great place to focus on that, so why not do that? I would go to the Loire FIRST and then finish in Paris and definitely no fewer than 5 nights for Paris which gives 4 full days and then you are there to fly out without wasting the penultimate day rushing back. The first day is a bit of a loss anyway with jet lag and all, so use it to get to your farthest point. Maybe train to Tours and then pick up a rental car the next day to tour the Loire (perhaps base in a small er town for the a couple of days and then drop the rental on your way into Paris. When we do this sort of thing we drop it at Orly and then cab to our lodging.

You don't have a lot of time which also suggests staying close to Paris to minimize travel.

And from Paris Versailles and Vaux le Vicomte are two 'must see' chateaux IMHO for chateaux lovers.
https://janettravels.wordpress.com/2010/09/20/vaux-le-vicomte-our-last-chateau/
Another idea to consider. Book an apartment for 8 or 9 days in Paris and then day trip to some of the many chateaux in the region. Here is Vaux le Vicomte

But there are also easily reached: Versailles, Fontainebleau, Malmaison, Sceaux, Ecouen, Rambouillet, Chantilly, and right on the metro the old fortress of Vincennes. And then you are in Paris for dinner and can get a comfortable base. It is really nice to have an apartment and be able to go out and get fancy Paris breakfast breads in the morning even if you don't cook dinners in. (there is also a lot of great take out available in Paris) You could even do a quick overnight or a day trip to Tours and do a one day chateaux tour from there. There might be bus tours for this out of Paris as well.

If it were me I would spend the time in Paris and do the day trips to chateaux (and the Navigo Decouverte travel pass will take you anywhere in the Ile de France and so to most of these.) I think Chantilly is outside the region although easily reached.

It really depends what your next stop is after France as to what is easiest.

Last summer, over 3 weeks, we visited Normandy, the Loire Valley, Provence and Paris. Based on that experience, I would suggest either Normandy or the Loire Valley due to proximity/reduced travel time to get there. We loved both areas. If you are obsessed with the chateaux, then go there. You can't go wrong, and what someone else would do is not important. We found Amboise a comfortable place to stay and, with a rental car, visited Chambord, Cherveney, and Chenonceau. We also visited Amboise as it was right there. I would return there in a heartbeat as it was beautiful, relaxing, very interesting, and easy to traverse.

Your OP doesn't mention if you can have a car in the Loire, which would be appropriate for 4 days. On a shorter visit, you could be satisfied with the limited chateau-shuttle services. Public transit is not really an option within the chateau zone. You can avoid a car more easily in Normandy, but it limits what you can see that way.

You don't say if you have a strong interest in wine. Do you want outdoor activities, like hiking and kyacks? Or do you want to see mainly uniquely French things? It sounds like museums outside Paris aren't important. Unlike many posters in this particular board, you don't say if you have an American-heritage interest in the Normandy beaches. Your home is not visible in your profile.

If you like Chateau,I would definitely go to the Loire Valley,we did it in September and stayed five nights in Vouvray in a B&B which is fifteen minutes from Amboise which is a great place to base yourself.Amboise has Clos Lucie and the castle and good restaurants.We did Chenonceau,ChaumontChaumont,Cheverny and Chambord.Azay Le Rideau was covered with scaffolding so we didn't visit it.
Mike

thanks for all the answers, i guess i'll stick with the Loire :)