Sorry it ended being too long ... but it is late, so I will let it be.
I used to be very fearful of talking, I would be ok in 1:1 or 1:2 scenarios. But put me in a bigger group, and I will not speak a word, Zero. And I would panic if the spotlight was thrown at me. I have spent entire business dinners (in group settings, then age 30 or so) without speaking. While I am still awkward at times, a lot of the fear has gone away. But it takes years (or decades).
I think the top comment (currently) by perardi has the right ideas, let me rephrase it to say: practice, experience and time. You can practice practice practice but you need real life experience, and sometime things just take time.
To begin with, I had the burden of stammering for the first 10 years of my life. People would constantly make fun of me. I eventually fixed it but it creates fear of speaking. I come from a country in the Asian continent, and did not speak English (could write) till I was almost in college. Being able to speak English took a few years to fix. I remember having to say "jealous" in a college setting, the first time I tried to speak it (had never heard someone verbalize it till then I think), and eventually I could not, so I spoke it as if I was spelling it. Laughter apart, the associated fears of people making fun never went away, and it became a part of my psyche. Then when I could speak without worrying about English failures, I will speak at the pace of a rocketship. Top speed only - always on Autobahn. Yikes.
Things are different now. I am not an extrovert, but I get commended for being able to explain things clearly (verbally). In my case, it was primarily the time that helped. I did not practice much, and experience came by being forced in situations. In hindsight, practice and (more) experience would have helped, but my fear would not let me. And for more than a decade I did not believe I could fix it because everyone else could communicate so much better than me, that I always thought I was different and this was my life !! I did read about the tricks of public speaking, but they only provided incremental improvements, never a sea change.
What eventually changed is 1) decades long time+experience followed by 2) me deciding to change it. So of course with time you get thrust into many opportunities for presentation and leading projects, maybe make some small talk, and each of those eventually helps. In the past I would have passed them on, but at some point I decided to take them on. Rather than me following a script at someone's direction for presentations / meetings, I used the opportunities to build my own scripts and the entire narrative. What helped was an understanding management who did not push me, so I was driven myself. And once I executed some of these on my own, the fear just went away. And once the fear goes away, it takes care of lots of other issues - it frees up your mind to think about the pace, think about the audience, think about the live tweaking of the content based on how the presentations are going - it is just amazing. Does not mean the nervousness has completely gone away, no, it is there, but you know that with enough preparation and rehearsals, and by knowing the content you can handle most things.
But this started by being "shy and socially awkward". What does presentation skills / public speaking have to do with that? My personal experience is that they are somehow tied together. I am not in a dating world trying to meet new people, but in a professional world, I am less wooden and able to relate to and converse with lot more people than before, because I know ... I can.
Does that mean everything is fixed ... I wish :)
In hindsight, if I had put in hard work to practice and gain experience (by finding opportunities), I think I would have shaved off 5-10 years of the additional time it took. Even if you (or I) are not capable, practice helps. Having the experience helps. And time helps because from each successful or failed experience, your body or mind has had the time to internalize these learnings.