We were in all of these regions last summer.
You don't say, but I assume that this 3rd region would be for only 2-3 days and that the total time for the trip is around 10 days. Is this correct?
If so, then time is your biggest enemy.
A good, basic & quick visit to the Dday area would be a 1/2 day from Bayeux (spend the other 1/2 day IN bayeux & sleep there). Visit omaha beach, where there is a small memorial and then just up the hill is the American Cemetary, where there is a very good museum that tells the story of the invasion in very human terms. So that's all one, very busy but productive day.
Then I would skip the Loire valley all together. Why? Because there are two reasons to be there; 1) to slowly absorb the lovely valley region, which you don't have time to do and 2) to see chateau that will all seem rather dull after seeing Versailles. The chateau are mildly interesting, but let's face it, there is something rather repugnant about that much wealth wasted on little-used piles of stone. So see the biggest pile while you are in Paris and skip the Loire because you don't have time to soak it in.
But I think it is important to spend a few days seeing rural France (to contrast with Paris), so I would spend them in the Dordogne river valley. Base either in Sarlat, which will have many options, or in one of the dozen small riverside villages a few miles south. Spend your days wandering these villages (each with its own castle or small chateau) and soak in the history, the truly genuine cuteness (as opposed to Disney-esque cuteness of the more touristy destinations), and the relative peace and quiet. It is a place worthy of remembering.
So maybe 4 days in Paris, 1 in Normandy, 4 in Dordogne and 1 back to Paris? Or 3-1-4-2 would work, too and you could sleep in two different neighborhoods in Paris to compare & contrast?