Re: approaching random girl at coffee shop?
Strange. I get this feeling you already know the answer so I'll try an answer something else. Hopefully it'll help.
You mentioned not wanting to seem awkward or "staged." Are you assuming that if you approach her that she would know you were up to something?
If so, then you're right. Most women assume that a man talking to them is hitting on them. Even my own gf assumes a guy asking her where a classroom is, is just trying to hit on her. (And she may be right) But if you get that eye contact with her that tells you she's interested, then she WANTS you to hit on her. Women are not enemies that need to be conquered. They want the same thing that we do. To meet someone interesting to spend time with and, if attracted, become sexual.
Took me awhile to see this myself. I felt that once they knew I was interested then the game was up. If this is your issue I could explain further. But I may be way off base also.
"All things are subject to interpretation. Whichever interpretation prevails at a given time is a function of power and not truth."