Booking hotels/hostels or not?

Usually you can still do it. There are exceptions to it like let's say Octoberfest in Munich when it's better to book well ahead. When I go to Europe I usually book first few days and then I am flexible.

Eight days total trip ? I would pre-book so my "finding a place to sleep" time would all be done at home rather than cutting into my vacation. Booking.com is fast and easy. But it's a personal preference thing.

Things have changed in the 40-50 years since your last trip. Internet has made booking ahead and route planning much easier and therefore people do it. Sure you can show w/o reservations and waste time attempting to find a decent place. You'll end up with whatever the other travelers didn't want. That it itself is reason enough for me to book ahead.

We lived in Germany and just jumped in the car and found hotels or B&Bs while traveling. That was in the 80s, light years in the past. We generally did fine, until on one trip, we arrived at our destination to discover every room in the city was booked due to an event. We had to move on to another location.

Now, I always book in advance. I plan the trip, transportation, most tours or DIY and especially hotel rooms.
The internet allows you to use competition to find the best price for best location with the best ratings. The beats the heck out of wandering on your own, hit or miss.

Pre book. Some places still have these offices but some have closed. Less demand because people can more easily do it themselves online. I am 36 and have never once used one (in the late 90s I would walk into hostels and after my hostel days were over the internet took over!)
and I don't know anyone who has recently. Obviously people still do - just fewer, so I would assume fewer hotels use the office and instead list online.
At the very least I would use wi-fi to book a day or two before you arrive in a city. Why risk not finding something? Waste of time when you can take care of it ahead.

You will need a phone if you are staying at apartments where you need to call the owner to let you in when you arrive. For hotels with a manned desk where you can walk in anytime they are open you just need internet.

I've made two summer-long play-it-by-ear trips in the last two years. I do not recommend depending on day-of arrangements through the local tourist office. In addition to the issues already raised: T.O.s used to be in train stations or just outside them. Now they can be a mile or more away, and you can no longer assume there will be lockers or a luggage-checking facility at every train (or bus) station.

I always want to find the T.O. to pick up maps, etc. I've encountered two T.O.s that were not in the places I expected them to be. I wasted a lot of time looking for the relocated T.O. in Berlin. That wasn't critical on such a long vacation but would have been massively frustrating on a short trip. Fortunately, I had already checked into my hotel, so I wasn't dragging luggage around with me.

Another thing to keep in mind is that in smaller cities the T.O. sometimes takes a mid-day break, though I don't know how common that is in Germany (as opposed to Italy, for example).

So, as others have said, it's far more prudent to arrange your lodgings a bit ahead of time via the internet. I do that in the evening when I'm relaxing back in my hotel, typically 24 to 48 hours in advance. I'm not picky about hotels, so it usually doesn't take terribly long to find an acceptable listing on booking.com, though it's common for the more conveniently located places to be booked up. (Checking those hotels' own websites may find rooms available.) I frequently have to stay in a nicer/more costly place than I would prefer, just to be sure I have air conditioning. The latter will presumably not be an issue for you in Germany in early May.

Note that I usually stay in each place at least three days. If you plan to move on every day, having to make lodging arrangements every evening may get tiresome. You'll also want reliable Wi-Fi in your room, and that means diligent reading of hotel reviews and elimination of places with multiple critical comments about Wi-Fi quality. Even doing that, you may occasionally end up in a place with a recently-broken router.

You can still book or check at the Tourist Office in Germany, as was done in yesteryear. I did the same in 1977. Still, doing it on the internet is preferable since it gives you more choice, and thereby, flexibility. Hostels do fill up, depending on the place and the particular hostel. If you go to towns that are frequented a lot by tourists, esp international tourists, then book ahead. If it is a town only Germans go to, you can still do the reservations the old fashion way..more likely. .

In 2007 I did that with a Pension in Münster/Westf., by calling them up two days in advance. I got a single I wanted, since I knew international tourists don't go to Münster as they would to Cologne. Keep in mind too that some establishments don't offer Wifi. Where I stay in Berlin, that Pension doesn't offer Wifi...makes no difference since I don't need it anyway.

Many thanks to all who have made suggestions and comments. You have all been so helpful! We will be booking ahead for sure, and I'll have my daughter set up phone service - it looks like we may need it. We'll probably be booking trains as well. I had thought about booking a train for just the first leg, but it seems sensible to do them all.

It's great to hear that you're going to pre-book accommodations. Although it shouldn't be as busy in April & May, some hotels could still be booked with tours or whatever. I prefer not to have to spend time wandering about in an unfamiliar place hauling all my gear and trying to find a place to sleep. Booking in advance gives me the choice of location, price and comfort level.

Gute Reise!