When I first started doing this, I found it very hard for similar reasons as you. What helped for me was to adopt the mindset of "playing a role" when doing it -- that the person shaking hands and meeting people wasn't me, it was a character. It really was me, of course. I didn't adopt a different personality. I'm just talking about taking an internal mental stance, a little self-deception.

This wasn't my idea, it came from the fact that so many actors on screen and stage are very introverted, but they don't they don't feel it so much when they're acting because the person on the stage isn't them anymore, it's the character.

The easiest way to get started is look around for others by themselves, then walk over and say hi. Realize they're likely feeling similar to you.

Every time you sit down for a talk, challenge yourself to sit near someone and introduce yourself. At lunch, find an empty seat at a busy table. As you do this, you'll see the people you met earlier and will feel more confident.

Also ask people if they know if anyone's gathering afterward. Nearly everyone will either invite you, tell you what they know, or be interested in doing something after too.