I've found a CBT* technique useful for overcoming that sort of anxiety (it's called catastrophising). I write down the situation and my prediction in a spreadsheet. An example would be: Situation - at work, I emailed [girl]; she hasn't emailed back yet. Prediction - She is going to break up with me.

Then when you receive an email back, you write down the outcome in a third column, e.g. received email back - we are meeting up tonight.

Looking back over the spreadsheet, you can see how accurate your predictions have been. I expect they tend to be too negative.

*CBT (Cognitive Behavioural Therapy) has a strong evidence base.

ETA: I hope that explanation is clear - I'm in a bit of a rush right now! I really wanted to explain it though, as it's had an extremely positive impact on my anxiety levels.

"Oh no, what if X happens, what'll I do? I might lose Y!"

I find that asking and answering a couple different question helps.

"What'll I do if I lose Y?" Not gnashing your teeth over losing Y, but actually answering the question. "If I lose Y I will ..."
"What's my best strategy here?"

One claim in some depression book - catastrophizing is at bottom a worry that you won't be able to handle the loss. It's a boogeyman in the closet, and you're too scared to open the closet and check, so live in fear, night after night, that the boogeyman will jump out and get you.

Open the closet and look. The boogeyman is there or he is not. Do you think he won't come out and kill you just because you refused to look to see if he was there? Buck up and look. What you'll likely find is that the boogeyman really isn't so fearsome, and your fear of the boogeyman is more crippling than anything he might do to you.

Either you'll spend from here to eternity with this fantastic girl, or you won't. Likely not. Someday you'll lose her. You'll weep and moan and move on to some other girl in a few weeks. Or maybe you'll dump her. You'll look back on your time together fondly, perhaps guiltily, when you're with someone else. Or perhaps 5 years from now you'll be struggling to remember her name.

Closet opened. Boogeyman faced. Situation handled.

Now, you look at your options in a game theoretic sense.

She's on the verge of dumping you, or not. You can fret about, or blithely go on as if nothing is wrong. Fretting makes either situation worse. Dominated strategy. Do not do.

You can handle it if she dumps you. Trying to read the tea leaves for the impending dump only makes you look and feel insecure. Do not. She'll eventually let you know if she wants to dump you. Until then, you're seeing a fantastic girl. Make hay while the sun is shining.

I'm a worrier. I'm trying to get over it. Part of it seems like fear is a response to uncertainty. Remove the uncertainty. Face the issue, pick a strategy, and move on.

Have you dated before?

It sounds like these contact points are natural accumulation points for your anxiety - not so much that you're specially anxious about them.

That said, what sorts of things are you requesting? I'd recommend a shift to make suggestions, or invitations where a 'no' answer can very clearly (even to you) be accompanied by 'It's sweet of you to offer, though' rather than 'GTFO'. If requesting something comes up so often that you've noticed the pattern, and they cause an aversion, perhaps you should avoid making so many requests per se?

You've already recognized the pattern. Now, when you send her an email, before anxiety kicks in, plan out a realistic anxiety schedule. You tell yourself, "I am sending this now, and I recognize that she might not respond tonight. Or tomorrow. But she usually replies within three days. If she hasn't written back in four days I'll be justified in feeling anxious." Precommit so you're not constantly thinking, "It's been 7 hours. That probably means it's anxiety time, right?"

I think that CBT techniques could be helpful. They tend to be about reasoning things out on paper in a structured manner.

You could work on realistically assessing how likely the feared result is, and on assuring yourself that, even if it happened, you could cope with it.

I'm resisting the urge just post a one line comment saying #humblebrags by posting the comment anyway and giving some advice :)

Below Joshua mentioned studying PUA and I agree. One method that works for me that I learned from PUA is to try to convince yourself that you don't care if she breaks up with you. Tell yourself "she's just a girl" or "there's plenty of fish in the sea" or whatever motivates you. I also combine this with a "just fuck it" attitude. If I find myself worrying about something like this, I'll say either in my head or out loud "fuck it. If she doesn't want to respond I don't care. I'm going to go play video games/exercise/whatever." (The "just fuck it" attitude only seems to work specifically with social interactions, in my experience. If I have reservations about other things, typically it's for a good reason)

Typical mind fallacy applies here - what works for me might not work for you or might have negative side effects, but you don't really know until you try.

But there's certainly no evidence supporting the idea that this is likely to happen, nor is the anxiety helping me prevent it or helping me in any other way.

A simple way to put that anxiety to use would be whenever you feel that anxiety, to do 10 pushups. Becoming more fit makes you more attractive, so it's not irrational to start doing more sport to increase your chances with woman.

Channel the anxiety into some useful activity.

From personal experience the best advice is to date a lot and get hurt a lot and build up a thick enough skin to where you don't care anymore about the rejections.

Worrying about the rejection will only make rejection more likely.

Act as if you are a confident person, then other people think you are confident, and you'll become more confident. While of course actually trying to do things to actually become more capable too, since that improves your confidence as well.

The other ideas here also are good techniques too, but what I found is that when I had been burned enough to stop caring about rejection was when I suddenly became successful at dating. The main thing that had changed was not worrying about it.