Thanks for the advice so far.
So let me just confirm this.
My visa runs out on the 27th of august.
If I were to go to Turkey or Egypt say on the 22nd and then return back to Germany on the 29th, I shouldn't have a problem?
Or at least to your best knowledge.
As that's what I'm planning with the advice you have given me.
I would not be so confident that the above will work. You will find that the German border/passport control people will look into your passport in great detail - I know this as my wife has a Ukranian passport, and they always look for very clear evidence that all is in order for her to come back home. And remember that if they are unhappy with the situation, you may find that you might not get back in at all.
In my opinion, there are two possibilities. Get official approval for an extension (via the university, Ausländerbohorde etc), or else return to the USA and then you can come back on a normal tourist visa.