Re ## 0, 2, 4, 6, 11-13.
I should have indicated that "What am I supposed to do?" (##0, 2, 6) is absolutely fine, as are some of the other suggestions--including #13.
My point (perhaps inartfully stated) in #11 was that "What should I do?" captures the meaning of the OP's German sentence and, to boot, is the most common way to express it--it's the most likely way it would be said (in terms of frequency, even though it's not necessarily better than the alternatives).
"What am I supposed to do?" (among other good choices mentioned above) is very common, but I think "What should I do?" is even more commonly used.
#4 correctly states: "An AE speaker would be highly unlikely to say 'what shall I do?'"
However, an AE speaker would be highly likely to say "What should I do?"
#12 asks how I would translate "was soll ich tun?"--without the "nur." I would translate it as "What should I do?"
As has been pointed out, the "nur" implies desperation here. However, I see nothing in "What am I supposed to do?" that suggests greater desperation than does "What should I do?"