Unfortunately, both Normandy and Verdun are at the other end of France from Nice, and transit will take some time.
For Normandy, you could fly from Nice to Lille, then rent a car. For Verdun, you could fly to Metz/Nancy, then rent a car. Driving all the way from Nice to these places will take hours; fine if you wanted to see destinations along the way, but not allowing you much time in the destinations themselves, before you have to be back near Paris for your flight home.
To investigate flight options for your dates, use Skyscanner: https://www.skyscanner.com/
A car rented in one place in France (say, Lille airport) and returned to another place in France (say, Charles de Gaulle airport) shouldn't have much, if any surcharge. You will need an International Drivers Permit, obtained at AAA, and carried with your US drivers license. You will also need to have a lot of cash for tolls (apparently acceptance of US credit cards is hit and miss), and will need to learn about French road rules and signs.
If you won't be driving, it's more complicated. From Paris, many base in Bayeux for a few days to take tours of the Normandy D-Day sights. I'm not sure about seeing Verdun without a car - even from Paris. For getting from Nice to Paris, the train and the plane are about a wash - by the time you add getting to and from the airports and waiting, neither one is a clear "winner" in time.