I totally understand your question. I had the exact same problem.
I also bought the stuff that is called brauner zucker at the grocery store and it wasn't what Americans called brown sugar. It was like what we call raw sugar or something. It was just brown tinted white sugar.
The closest thing I found in Germany to American brown sugar was the Bio Moscabo. It is organic unrefined full raw sugar. However, it has a very strong flavor that I don't remember the American brown sugar having. I made chocolate chip cookies out of it, but they had a different flavor. They were good though. But I still am searching to find something more like American brown sugar. I suppose maybe I should just use the "brauner zucker" that I originally bought and see how that turns out like elfenstar suggested.
But if you want to buy the Moscabo sugar. You can find it at most of the Reformhaus (organic kind of grocery stores). It is made by a brand called Gepa which is a fair trade product line. I also found it at Hertie's.
I've pasted in a picture of the package.
I ended up just buying both the light and dark brown sugars in the U.S. at Christmas time and bringing it back with me. Eventually my supply will run out, but maybe just before I go back to the U.S. again and stock up again! It's nice to have some things from home.