I'm French. Early June is the BEST time, OP! Weather isn't too hot, and school isn't out yet.
The Loire chateaux are beautiful. If you go down there, I recommend this place:
https://www.suitesdecheverny.fr/en/
If you have not gone to Mont St Michel, please do so. It's breathtaking, and one of my favorite spots in France. You can pair it with Normandy beaches with a little extra driving (there's a bumpy but highly informative WWII-era Jeep tour you can book for the landing beaches). If you decide on Mont St Michel, the best way to experience it is by sleeping on the Mont and waking up, opening your casement window, and breathing in the fresh morning air, then going down to breakfast. I recommend this hotel, but they fill up months in advance, so don't delay in making reservations:
https://www.auberge-saint-pierre.fr/en/