Man, there is so much to say, I don't even know if I'll make sense because of all these ideas in my head. Expect lots of things added in brackets. Either way, let's get to it

First off - Getting over approach anxiety
I prefer to do day game, so this whole post will be exclusively about what I do in the field during the day. I rarely have days when I go out to pick up, I just start conversations as i go about my day(that takes out a part the stress of approaching, like alleviating the fear of going home "empty handed". I have stuff to do and the girls i approach are a bonus to my day)
Although sometimes you don't feel at your best. We've all had some of those days. A good one in that would be some really easy warm-ups. Start conversations with people that are walking by. Something along the lines of:
"Excuse me, do you know what the time is?"
It doesn't need to be a number close approach. Sometimes we all need time to adjust and get into the mood.
The purpose of it is to just get the feel for stopping a person and talking to them without wanting anything physical from them. It's especially useful in the very early stages of practicing your game, since you are gradually getting used to human contact and approach anxiety pretty much dissipates.
On a bad day I approach about 3-4 people as a warm up before I am ready to go at the hottest of the HBs.

Approaching
Approaching is actually very easy once you are properly warmed up. Personally, I approach with situational openers when I feel warmed up. I feel in the moment and it becomes effortless. So something like:
"Oh, hey there. This may seem a bit weird, but I really like your shoes. Where did you get those?"
and then follow up with a little story to make my remark seem reasonable rather than creepy. For example:
"My sister's birthday is coming up and she is a huge shopoholic. So I figured I'd buy her a present she will thoroughly enjoy"
(This is basically DHV as well. It came out like that by accident, my bad)
After that the girl will most likely tell me about the shoes and where she bought them. From there you can introduce yourself, game naturally and change the subjects as you wish until you get to the closing. I generally talk in a "semi-direct" manner (saying things like "I know you are with me just for the sex " or "you will totally be my girlfriend, you just don't know it yet" or "even after all this time we've been together, you are still so damn attractive to me" for a joke) - hence I am completely honest (the girl understands it, and goes along) and both me and the girl have fun

Closing
Closing is really simple as well. I prefer to joke here and there to brighten stuff up a bit (as you may have noticed in the piece before), so I often end with something like: "I have to go now, but the chemistry we have is obvious. I think a girl like you deserves the finest. What do you say we meet again for a romantic date? Just imagine... You... Me... The free water at McDonald's... Maybe even with ice. I would like to spoil you a little" then do a seductive wink at her and it is set. haha. Actually, looking back at the past year that I used that line it has never failed me. Sometimes, though, I went on an instant date with the girl rather than get her number and say goodbye (If I'm not out with friends, I prefer the instant dates)

I didn't really know where to add this, but it has to be said:
Don't worry about the outcome. It's really not going to get any better if you are going to be really nervous while in a set. Everyone gets rejected and failure is just a hands on way to learn. Take out things for you to fix in your game. Just keep going out and keep trying. That's how the now near legendary Mystery learned, that's how Style learned. Hell, that's how we all learn. Some may take a little longer to get there, but the aim is similar for all of us. Don't let anything hold you back in your journey to becoming a better self.

Hopefully this helps some and gives some others food for thought.
Faithfully yours and stuff

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The one who conquers others is strong,but the one who conquers himself is mighty.