Not going to sugarcoat it - approach anxiety is a real thing most of us have gone through, you gotta face that fear and do it when it's nerve-wracking instead of thinking there will be a way to make yourself not nervous. I have two friend who are highly accomplished players who tell me they still get very nervous, but have personal "rules" to push through it and approach. Their results speak for themselves.
One thing to keep in mind here is that lots of people are boring, and that can especially be true for attractive women who are not accustomed to providing conversational energy themselves (people are always interested in them so others provide the energy). It's easy to beat yourself up about not keeping a conversation going, but you can't control her response, you can only give her an opportunity to respond to you; in time your mindset will shift to "pfffft, this woman is passing on her chance to chat with an interesting attractive guy."
One thing to keep in mind here is that lots of people are boring, and that can especially be true for attractive women who are not accustomed to providing conversational energy themselves (people are always interested in them so others provide the energy). It's easy to beat yourself up about not keeping a conversation going, but you can't control her response, you can only give her an opportunity to respond to you; in time your mindset will shift to "pfffft, this woman is passing on her chance to chat with an interesting attractive guy."