I think your translations are fine. "What am I to do?" sounds like old-fashioned American movie English to me, where the female protagonist finds herself in a sticky situation and asks herself rhetorically, "What am I to do?" Aside from your two suggestions, the other possibility could be ¿Qué haré?
Because I believe it's rhetorical question, I don't think ¿Qué tengo que hacer? or ¿qué se supone que tengo que hacer? would be appropriate, because it's too literal. The person saying "What am I to do?" is expressing dismay and confusion over their situation, not asking for a step-by-step list of what to do.
I like "¿Qué hago?" "¿Qué debo hacer?" o "¿Qué haré?"