Coming back to this now that I have more time. I have a number of thoughts.
1) Why Hannover? If you are visiting Amsterdam first, it's probably where you are flying into/out of, and as it is a major hub, that makes sense and likely makes it cheaper. So then you can train to anywhere. You don't need a true base if you are staying a few nights in each. So why stick to Hannover?
2) If you do prefer a base or there are flight reasons for choosing one city, consider Hamburg. It is likely easier to get in and out of. Bremen is pretty easy, too--and VERY pretty and interesting. I like it better than Hannover. Bremen is WELL worth a visit regardless, and it has a very different feel from the half-timbered stuff you are worried about seeing too much of. I would consider dropping either Wolfenbüttel or Hildesheim in favour of Bremen.
With that in mind, here are a few thoughts of places in northern Germany that may or may not fit your style of travel. I have written enough about Hamburg itself, so I will leave it off for now.
--Note that in the north, "half-timbered" is almost always brick, sometimes very beautifully done, and the timbering more square. You might also see a lot more reed roofed buildings. You would see this in places like Lüneburg and particularly in the smaller villages. In places like Goslar and even to some extent in Celle, you would see more of the plaster style with slate roofing. In fact, there are all kinds of architectural and regional variations within the category of "half-timbered" that could be a fun exploratory sub-theme of the trip. Lüneburg would also give you the Hanseatic style brick waterfront houses.
Verden is a surprisingly wonderful and undiscovered town near Bremen. It is a horse town, home of the German Horse museum--and even if you aren't a horse person, it could be neat. The horse has shaped human history around the world, and the Germans have been a major horse breeding country for hundreds of years. You may or may not know that about 30% of ALL the Olympic horses in the last 30 years are German bred, for example (as far back as I could find stats). The museum is also partially interactive, with riding and carriage driving simulators and more. The town has a lovely brick-and-half-timbered old town, a local museum, church, and of course farms and hiking trails to explore. Worth a day trip for most people, especially from Bremen, but probably not worth spending a night. But worth mentioning all the same.
If you like nature and being very rural (I know, you mentioned mid-sized cities), then one of the prettiest places I think I have ever seen is the Niedersächsische Elbtalaue. We had to pick up a vehicle in a "town" called Trebel once, and driving there was absolutely stunning. And took forever because, well, there is NOTHING but pretty nature. I don't know, though, if this is something for you without a car. It is pretty hard to get to / around. But better to mention it than not, and you might find a neat little farmhouse to stay in or something.
If you are venturing further north to Lübeck, the flavour of your trip changes as well. I like the suggestion. The Hanseatic element means you aren't just looking at half-timbered houses. Good call to add it.
If you are going to the Harz, you might consider Wernigerode. But the suggestion of Braunlage and Schierke was a good one. I have family with a vacation apartment in Braunlage, and they are there a lot--for hiking, but also because the town is adorable. Same with Schierke. Comparing the two is also interesting because they were on opposite sides of the former border. The Brocken, the highest mountain in northern Germany had a "weather station" on it that was absolutely for sure not anything else and definitely not at all used for spy work in the GDR times. And the narrow gage train is the best way to get up there!