Plan as we go?

What time of year? Is "we" a couple or a family with kids that will need a triple or quad room? Will you be driving or using trains/buses? How tight is your budget?

Sorry for leaving out those details! "We" is a couple and we will be traveling at the end of May, 2019. Our budget is not tight but not overly extravagant either, I'm thinking $250 or less per night but could be persuaded up to $300 per night if it was for something amazing. Also, we would prefer train but could rent a car if needed.

Would we be crazy to wait until we get to France to see if we are enjoying the bustling city or the quiet countryside more and then try to book hotels, etc.?

In this age of mobile technology where you can book things on the fly on apps as long as you have wifi it is not crazy. Travelers that do not need everything planned out I think would like to that kind of option or freedom. However the stuff (e.g. trains or hotels etc.) where you need reservations can cost more at the last minute; nonetheless maybe you have the means to cover that possibility.

That said, I once bought a one way train ticket in May 2002 to Belgium from London on a whim about an hour or less before departure paying the max but I did not care.

Having a car really gave us the freedom to visit D-Day landing sights, Bayeux (where we stayed one night), and the museum in Caen on our own pace. Late May isn’t the very height of vacation time in France, so winging it as you go would work for lodging, and things shouldn’t be already booked up. Are you visiting St. Malo and Cancale, I hope?

You can improvise as you go as long as you are not picky about where you sleep. Late May can be pretty busy in some of the popular places.

But I would not say "zero planning," rather flesh out the possibilities before you go so that you can make an informed choice.

Also, if you are in Normandy with a flight home in 3 days, your options are pretty limited. It's basically a full day in the Loire or getting into Brittainy, then back to Paris to spend the night and catch your flight.

You might have more freedom if you see Normandy first.

We have done this many times in France, even during the height of summer. Go for it.
Booking.com is your friend...makes it very easy to decide where you want to stay at the last minute as long as you're not terribly picky!

Based on the information you've provided, I don't see any reason to discourage you, but I recommend that you Google for special events in the area so you'll know ahead of time if you may have difficulty in a particular spot. I didn't do a good enough job of than in 2017 and had a lot of trouble finding an affordable room in Provence, even though I booked several days in advance.

Do you mean three days elsewhere in France outside of Normandy, for a total of eight days? If you think you might choose Paris for those three days, I would not do it on the fly. In the countryside you could do this with your budget. The thing about being able to book on the internet cuts both ways. It makes it easier to book on the fly, sure. But because everyone else is booking in advance because it's so easy to do on the internet, all the best places are taken. You will end up compromising for location, quality, price, etc. But if you are willing to spend $250-300 per night you would probably be fine. (Although if you are willing to pay that much for countryside or smaller city accommodations your standards may be way higher than mine! I target 8+ ratings on booking.com). It's a high price to pay for flexibility, when France outside of Paris can otherwise be a bargain. But if you prefer flexibility over saving hundreds of dollars for those three days, it's your trip so go for it. You should decide the rental car thing in advance though, unless money is really no object.

we will be traveling at the end of May, 2019

This is a big red flag if your interest in Normandy is the D-day landing areas. Late May will be bumping right into the 75th anniversary of D-day. Many places in that area are already booked, and have been for quite a while. During the 70th anniversary, I know many people who stayed well outside of the D-day area because that was all they could find for availability. If this is the area you are interested in, I'd make firm plans and bookings now if you can. If you are looking at areas in Normandy that are far from there, you may be able to wing it.

Thank you all so much, this has really helped. Thank you to the person that mentioned St. Malo, that was not on our list before but definitely is now and may be where we spend the last 3 days, meaning we can book in advance. Thanks again, everyone!

I sometimes book ahead and sometimes book on the go. In May you won't go homeless. My former French girlfriend normally booked last minute hotel rooms using the trip adviser app on her phone. Sometimes I follow her lead and have had good results.

The only place I've paid close to $250 is in Paris, where the nice hotels can be very expensive. Usually I pay around 100-135 euro per night in Paris for a double room that is decent but not luxurious (Ibis business hotel or similar). I usually travel spring and fall, avoiding July and August which are likely the most expensive months.

Elsewhere around France I've rented nice places, mostly through Airbnb, for closer to a 50 euro average. A lot depends on what you want and where you're looking but May isn't an expensive time of year.