We always rent a car in France. Even without gps, signage and directions are easy. Before leaving, I would check out driving time and distances on viamichelin.com. to help plan your route. Just enter departure point and destination, and it will give you options for your route, driving times, etc. I planned a three week driving tour of northern France using the site, so we knew how much drive time was needed on driving days and we knew if we had planned too much driving that day, and could alter our destinations accordingly. It is user friendly.
And yes, having a car in Normandy is very useful and gives you much more flexibility in deciding what to see and where to go. There is so much more to see and do there than just beaches and rocks. The British and American museums there, the American cemetery (especially at flag-lowering time), the Bayeux tapestries, the great ciders to taste and meals to eat (shellfish, lamb dishes, buttery pastries...yum!) and the beautiful cities and towns with histories that go back MUCH farther than WW II.
Do remember that Strasbourg and Brussels are quite a way from Normandy. I hope you don't shortchange your time in Normandy. You will need two full days to see the highlight sights of WW II. Give yourself at least another full day to visit the Bayeux tapestries (which illustrate the Battle of Hastings in 1066) and to wander around town. Bayeux is a good town to use as a base if you don't have another place in mind.
Bon Voyage!