A one-day small-group tour with a company like Overlord will give you a decent overview of some of the key invasion sights. There are lots of potential uses for an extra full day or two or three:
Bayeux itself has the tapestry, the cathedral, the historic center, the British cemetery and the nice invasion museum. Due to the variety of sights and the fact that most of the D-Day tours originate in Bayeux, it is usually the best base for folks who don't want to drive themselves around the area.
There are some invasion-related sites accessible by public transportation. I found the museum in Falaise focusing on the civilian experience during the war (which also covers the Resistance) quite interesting; there's bus service from Caen to Falaise, and Caen is a short train ride from Bayeux.
Caen was very heavily bombed during the war and doesn't have much historical architecture left. There are a couple of abbeys plus the castle. The biggest local tourist destination is the Memorial de Caen, a history museum than can fill a full day. I liked it a lot, but many folks find it overwhelmingly large, crowded and expensive. It covers the lead-up period to WWII and the Cold War as well as the war itself. Many people apparently go to the museum interested only in the D-Day invasion; those folks are better off choosing a different museum (such as the one in Bayeux).
There are attractive towns scattered along the coast east of Bayeux: Cabourg, Deauville/Trouville, Honfleur, Etretat. Honfleur is the one most often mentioned on this forum. Some of those towns are reachable by train; others--including Honfleur--require a bus for at least part of the trip. Caen is a more convenient base than Bayeux if you want to spend more than one day seeing such places.
Rouen has a lovely historic center that was rebuilt (apparently very accurately) after wartime destruction.
As you can see, there's plenty to do in just those parts of Normandy to fill a week or more. It all depends on what you want to see and how much time you have. Including Alsace (also a great destination) will cut into the time you might otherwise spend in Normandy and the Loire Valley. Note: It is highly unlikely that spending three nights in Normandy will yield four days of sightseeing time. I would count three nights as two days. It will take at least half a day to travel to and from Normandy and get settled in your new lodgings.
Normandy and Brittany are typically rather cool and overcast even in mid-summer. I'd expect it to be even more so in April. The Wikipedia entry for Caen has a weather-summary chart that will give you an idea: https://en.wikipedia.org/wiki/Caen. As you can see, the average high for the month is 56.5F and it typically rains on one day out of three, but the amount of total rainfall is not all that high. Unfortunately, those figures are based on an historical period that ends in 2010; weather seems to have been getting wackier and wackier. When considering the timing of my trips, I like to have more recent and more detailed information. I go to timeanddate.com for actual, historical, day-by-day weather data so I can get a better idea of the range of conditions I'm likely to encounter at the time of my trip. That link will take you to Bayeux's statistics for April 2022. I always check at least the most recent three years' data; five would be better, because there can be a lot of variation from year to year.
I haven't been to the Loire Valley and cannot comment on the amount of time you'd want to have for that area.
