I've talked to a few neurotypicals about this kind of thing. I'm told that everyone feels anxious about meeting with someone they're attracted to. Some just hide it better.
On the rare occasions I can get a girl to go out with me, I'm also super nervous. I've learned to enjoy the anxiety, though, if that makes any sense. Like going on a roller coaster; it wouldn't be as fun if it wasn't a little intimidating.
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"I'm a weird dude, and it doesn't always work out in my favor, but the whole thing is supposed to say something: That a weak dude like me, odds against them, can make it, so can everyone else."
-Aleksander Vinter, aka Savant, EDM Producer