OP, it seems mainly like you need a good confidence boost to get you approaching. There's tons of write ups about developing self confidence for game and for other reasons, and I"d encourage you to find a gym or take up some other exercise program to make you happier and more confident.
All that being said, if you really need a springboard to get past your self doubt and move on to approaching, I can recommend a few things you can do that might help you out.
The first is, if you're really weak socially like I was for a very long time, just start talking to people. No flirting, no romantic pressure, just get used to striking up conversations with strangers you meet in the day to day. While this doesn't directly impact approach anxiety, if you have anxiety in social situations in general this will help you get past that. You'll probably even make a few friends doing this.
More specifically for approach anxiety, I encourage you to do the following. Sit down with a pen and paper (or a document on your PC, whichever you prefer), and just let your imagination run wild. Think through all the negative things that could happen to you when you approach a girl, and no matter how ridiculous or trivial they are, write them all down. Then head off and do something else for a little while, and come back and read through the list. What you'll find is that almost everything on the list will either be so ridiculous it's laughable, or so trivial that there's no reason to worry about it in the first place. Then take the time to reflect on all the good that can come out of it, and recognize that the possible good far outweighs the possible bad. This should help you springboard yourself into make a few intiial approaches, that will hopefully lead to more and more as you get more used to it.
All that being said, if you really need a springboard to get past your self doubt and move on to approaching, I can recommend a few things you can do that might help you out.
The first is, if you're really weak socially like I was for a very long time, just start talking to people. No flirting, no romantic pressure, just get used to striking up conversations with strangers you meet in the day to day. While this doesn't directly impact approach anxiety, if you have anxiety in social situations in general this will help you get past that. You'll probably even make a few friends doing this.
More specifically for approach anxiety, I encourage you to do the following. Sit down with a pen and paper (or a document on your PC, whichever you prefer), and just let your imagination run wild. Think through all the negative things that could happen to you when you approach a girl, and no matter how ridiculous or trivial they are, write them all down. Then head off and do something else for a little while, and come back and read through the list. What you'll find is that almost everything on the list will either be so ridiculous it's laughable, or so trivial that there's no reason to worry about it in the first place. Then take the time to reflect on all the good that can come out of it, and recognize that the possible good far outweighs the possible bad. This should help you springboard yourself into make a few intiial approaches, that will hopefully lead to more and more as you get more used to it.