I actually don't get a lot of anxiety when I public speak for some odd reason, but I do when I perform music in public. They are a bit different, but they are mostly similar in the sense that we have to do something in front of people and it scares the crap out of most people. So here's what I do when I get nerves, but I will translate most of it into speaking terms rather than musical terms.
So what to do before the big presentation? Practice, practice, practice. But not too much or else you might start to mess up. Although be prepared to what you want to say. Rehearse it in your mind and make your points solid. Don't memorize word for word what you're going to say. Instead, think of the main ideas and if you are prepared, what you'll say will often come naturally. Just be sure to study your material well, inside and out.
What do you do once it's the big day? Best thing to do is to find your center before you go up. Start off by picking a focal point or a fixed object somewhere off in the distance that is below eye level. Having your eyes look below helps you with concentrating and blocks off distractions. Then think in your mind what you want to achieve. "I want to make the best speech ever" or "I'm going to persuade the judges or my opponents that my point is right (if you are debating)" are just a few examples. When you make these goals in your mind, use positive words or phrases like "I'm going to...", "I want to...", "I will...", or "I hope..." and avoid using "don't do...", "try not to do...", or anything negative. For instance, you want to make your speech interesting. Don't say "don't make my speech boring" but instead go "I'm going to make this speech awesome!" Also, beforehand, try to prep yourself with words that will trigger sensations or feelings. You'll have to do this weeks before you actually go up or else it won't work (yes, I am implying that you practice centering even when you are simply practicing your speech). Like maybe if you think of the word "steady", the feeling you get is a confident sounding voice that projects to the audience. So bring up those triggering words again into your mind and think about them. As you are giving yourself these mental preparations, physically prepare yourself as well. Being anxious causes your muscles to tense up so lose them up by shaking them out, squeezing them really hard and then letting them go, or stretching them out. Then try finding your center where all that energy and stress is build up. I usually find it just right above my stomach. Focusing on that usually helps you calm yourself and blocks you from distractions. Then when you are in touch with your center, try to imagine yourself redirecting all that energy around you to your whole body. Think of the energy as going to the edges of your body and then beaming out of you like a radiant of confidence and persuasion. I know, it sounds bizarre, but it works. And as you're doing this, remember to breathe properly and efficiently to get enough oxygen in your brain. When we are nervous, we tend to take shallow breathes and our brain doesn't get enough oxygen, causing a blurriness in thought. That's why you need to take deep breaths through your nose and exhale through your mouth very slowly. Doing this technique at first will take a lot of practice and you may need to prepare yourself half an hour before your presentation. But once you nail it, you'll be able to do it just a few minutes (or even less than a minute for some) before you need to talk. This practice was developed in the 60s or 70s I believe for athletes. Now musicians use this technique as well and I think that it can work well for public speakers too.
Then as you are making your speech, remember to keep your thoughts positive. Don't put yourself down for making that one slip up because this could affect your entire speech. Just go with it. Improvise if you need to. As long as you have your materials solid, improvising a few things shouldn't be as scary as you might think. You may not have delivered it in exactly the same way you had wanted it, but at least you'll know what message to deliver and won't stand there looking like a lost sheep. It also helps to maybe squeeze your toes as you are talking so then some of that anxiety energy can transfer down to your feet and be released, driving it away from your head and torso so you can think clearly. Then lastly, don't think about what other people are going to think of you. Think about what you want to do it and try to make that a reality, regardless of what the audience may think. Even though in the long run (especially if it's a debate) the opinions of others will count, just try not to make it count for that moment when you are on the spotlight. Be passionate about you're saying and make it relate to you. When you're talking, it's all about you so make it that speech all about you.
So like I said before, I don't know if this will specifically work for public speaking but this is my best advice for when you're performing. And I believe that performing and public speaking share a lot of similarities, so I wouldn't be surprised if it actually did work for public speaking.
