Whether it is Füssen, Garmisch, Oberstdorf or any of the other villages of the area actually doesn't matter too much if you are free in your choices. If you plan a stay at Füssen anyway,it's like mvs says: you can remain there for the whole stay and have good choices. In fact, Füssen is rather centtral - you can reach several mountain groups within less than an hour:
If you look at the maps on the pages you won''t find Kempten - but it's always just outside the frame. For Wetterstein, it''s half a frame to the left, for the Lechtal Alps, it's some 10 miles north of Reute, for the Allgäu Alps it's just outside the top right corner and for the Ammergau Alps it's left of the large lake you can see there.
Generally the Alps are crisscrossed by hiking trails - marked hiking trails that is - so you won't have trouble finding an appropriate trail for you and your wife. Probably weather is a more severe limitation as there still might be snow on the upper north sides of the mountains. Get yourself a good map - Kompass does some good ones - most of them 1:50000 which would suit you well. You'll get the maps for between 6€ and 8€ at any bookshop in the area, sometimes even in the supermarkets. All real trails are marked on these maps and you only have to take care not to get onto dotted ones or ones marked with a ladder. Those will probably exceed class 2.
As for suggestions: if you really want to see Neuschwanstein (it beats me how anyone might be interested in this fake castle) you can add Zugspitze, Germany's highest mountain, to your list. The crowds will be similar as several cable-cars and a train lead up to the summit. You'll probably even meet people you met in the queues at Neuschwanstein. However, there's not much walking area up there and at nearly 3000m there is bound to be snow up there, too.
If you don't want to reach the summits but rather also like the valleys, Lechtal west of Reute, is a good destination. It is rather lonesome (if there is such a thing in the Alps) and rather beautiful. The mountains might be a bit tough for your wife, but then, there certainly are the lower ones to be considered as well.
As for huts - most of them will be closed during the time you plan to stay. The winter season ends at Easter and the summer season starts in late May at the earliest. Some of the lower alms and huts might be open all year but they often don't take guests over night. Some of the higher huts have a "winterraum", which you can use, but there won't be any food and these rooms are rather only shelters.
If you want to leave Füssen for another place, I wouldn't go to Garmisch because it is too close. Both towns can be used as base for the same area. I would rather go to Sonthofen or Oberstdorf in the west, where you have a completely different lookout. And they are closer too Freiburg, which probably will be quite a hassle to get to from any of the towns close to the Alps. It's either a large detour along the motorways or a slow drive along small highways.