I identify with the feeling that you are kind of split off from yourself, watching yourself and making critical comments that induce more self-consciousness. When I was younger, it wasn't even people I was scared of - I felt foolish even in front of myself when I had to practice doing things that were not within my normal range of comfort, because I was something of a perfectionist. I didn't want anyone to see anything but the best version of what I could do before they critiqued it (I wasn't so much afraid of the critique as I was concerned about them seeing the best version before they commented) and I'd put off perfecting that because it made me feel so inwardly uncomfortable.
I love speaking in public now, but that was not always the case. Here are a few things I've discovered are helpful:
1) Familiarity with what to expect brings a high comfort level. We are generally afraid of being surprised by something and looking foolish in front of people we want to impress. Doing anything well requires practice to become proficient at it. It is easy to make excuses in your own mind about how it just comes naturally to some types and not others. Some types may be less likely to avoid certain opportunities so they have gotten more practice in, but it is not that anyone naturally knows what to do in every situation. Practice is what makes the difference.
If you have to speak publicly, you should have given several trial runs in a less threatening environment so that you learn what to do if you unexpectedly run into trouble and so that your adrenaline level is reduced to what is useful instead of what will induce heart failure! Do not make your main speech the only trial run you have. The more you know what to expect, the more comfortable you will feel. If you can, practice once in the venue you will be speaking in.
2) Your audience will notice and judge you much more on how you recover from your mistakes than whether or not you make them. It is very likely that you will make some mistakes. Make sure you have an idea of how you will deal with it so that your audience feels that you are in charge and will be just fine.
3) Know your material, and find a way to link it to yourself or others so that you are enthusiastic about it. Allow your passion about what you have to tell them to outshine the fear you feel in delivering it. People will pick up on your excitement and passion, even if they don't start out excited themselves. By knowing your material well, you not only reassure yourself, but you also then have room left in your head to focus on your audience and making this a fun or useful experience for them.
4) Practice speaking slowly enough and enunciate carefully. The number one thing that makes listening to a speaker unpleasant is when you cannot clearly understand them. The ends of sentences and words particularly tend to get thrown away. By enunciating well, you do not have to "turn up the volume" nearly as loud. Highlight the areas or words in what you are saying that need special emphasis and practice delivering it that way so that your audience can readily pick up your meaning. If you state something important that they need to remember (like a point in an outline), pause slightly to give them a chance to absorb what you are saying. You have a paper in front of you and know what you are going to say. They don't and need time for it to sink in before hearing you expand on it.
5) Find concrete, tangible elements for your delivery to focus your attention on. You should have several default areas/people in your audience to make eye contact with and try to project your voice towards. This ensures that you are not ignoring anyone, and allows you something specific to do that will calm your nerves. Skylight's suggestion about breathing from your diaphragm will help ensure that your voice is not strained and will also allow you to focus your attention on something tangible. Find out what your nervous tics are and replace them with something less distracting to your audience.
6) Look for low stress situations where you can become comfortable performing or speaking. This may be in the context of a larger group (like a choir, band, sports group), it may be doing some kind of teaching, it may be volunteering somewhere, it may be speaking on a subject at a kids group or community event, it may be looking for different forms of service towards other people (older people, people in hospital etc) reading to them and so on. It would be especially good to seek out environments that are low stress, but unfamiliar to you.
For me, over the years I've become much more comfortable interacting with old people, people with mental health issues, physically or mentally challenged people, immigrants or non-English speakers, people from other cultures and so on. I've realized that the common thread is learning to take the focus off of myself and focus on what the other person's needs are. If I'm not sure how to best interact with them, I've found that it's often useful to ask! It's awkward at first to know how loud to talk to a hard of hearing older person, or what to do if you don't understand something a non-native speaker says even after they've repeated themselves. It's hard to know what to do when you encounter a child in a wheelchair with profound cerebral palsy who cannot speak. Do you deal with it like a person who is mute? Like a baby? It's only exposure and asking questions that allow you to not feel or look foolish. Chances are if you feel awkward, other people also avoid those people because they are thinking of themselves first and don't know what to do with those feelings.
This public speaking thing is a great catalyst for considering how you go about your daily life in terms of developing the confidence and range of experience to put your focus on your audience instead of your feelings about addressing your audience, no matter the context!