Unless you are doing organized tours each day in Normandy, you will need a car to get around. Janettravels has a very good point about Paris. Have you been there? If not, of your 12 nights in France I personally would spend more time in Paris. There is so much to see and do there. You would still have more than enough time to spend in Normandy. What time of year will you be traveling?
I would structure my trip something like this-
Arrive in Paris in the morning. If traveling Spring - Fall, take the train to Giverny. Tour Monet's home and garden. Spend a night there to help with your jetlag. If traveling during the time of year that Giverny is closed, take the train to Rouen instead. Spend the night there. Pick up a car there for the rest of the time you are outside of Paris. With your allowable time I would limit myself to no more than 4 nights in Normandy. Bayeux is a nice place to base yourself. Are you interested in the Loire? If yes, spend 3 nights there. Again, a car is necessary. That's 7 nights. The remaining 5 nights I would spend in Paris. If you aren't interested in anything besides Normandy and Paris, I would allot 4-5 nights in Normandy and the rest in Paris.
As for jetlag, I've found that the best way to deal with it is to just keep moving. Being outdoors really helps. Stay up until 9:00 p.m. or so that first night, get a good nights sleep, then the next day just hit the ground running. You mentioned going to Edmonds, so I'll assume you live in Washington. Going to Europe is a long trek for those of us on the West Coast.
Have a great trip!