It sounds like you and she have fundamentally different personality types.
You sound like an extrovert, and like someone who wears her heart on her sleeve. The impression I get from your post is that the last thing in the world you'd ever want is to keep your feelings bottled up inside. You sound expressive, and that's a wonderful thing.
And, it also sounds like you're the mom of an introvert. Your daughter sounds like someone who processes things internally. I don't know what it's like for her, but for me, being compelled to talk about my feelings feels almost violent -- like I'm being forced to take what's most personal and private and sensitive and reveal it for the world to see. I'm also very uncomfortable with physical contact when I'm not happy. I'll hug and all that when I'm in a good mood, not when I'm not. This is every bit as wonderful of a way to be.
So, maybe the best way to help your daughter is to be present, to be receptive, but not assume she needs help. If she seems nervous, okay fine, she's processing her nervousness. Physical contact, talking it out, will probably just make things worse.
In terms of talking to other people, you might just help her make a distinction between her internal feelings and external expression, say things like, "You don't have to like people or talk to them for a long time, but it's important to be polite, so just quickly say hi." As a shy mom of a shy son, this is what I've done, and unless people are super-overbearing, it usually works well.
ETA: Reading this, I fear I may have been too harsh or something. You do sound like you understand your daughter's fundamental personality type and like you're invested in giving her the space to be who she is. I just want to encourage you even more in this direction, and to urge you to recognize that she may never, ever want to talk about her feelings, and that that's fine. But you do sound like a loving, tuned-in mom.