I used to struggle with social anxiety quite a lot, chiefly when having to interact with strangers. Over time, I've gotten to where I can speak freely and confidently with groups of maybe up to six to ten loose acquaintances at a time? Were they all people I knew well, I could theoretically address even more.
All bets are off, though, for very large crowds; I had to give a speech in front of my whole school once, and while I apparently did a good job with it, I was so thoroughly depersonalized that I barely remember it at all. Cold-calling strangers on the phone is still rather difficult for me to do as well, no matter how many times I assure myself that the conversation will be professionally civil at worst.
As for advice, perhaps it'd help to think of socializing as just a bunch of rules that you have to memorize and then follow when they become relevant to the situation at hand. It's less complicated than it seems; for an analogy, think of a social interaction as writing a letter. You usually start and end it with certain phrases that are implicitly understood as being the "appropriate" things to say ("Dear Such-and-such" --> "Yours Truly, So-and-so" vs "Hello, how are you?" --> "See you later!"). In between those two, you can say more or less whatever you want/need to say.
If the interaction calls for being polite, then make sure to sprinkle some "please"s and "thank you"s into the mix. Additionally, you could couch whatever it is that you want as being more of a question or hypothetical than a command (e.g. "Would I be able to take off this Friday?" instead of "I want to have a day off this Friday."), since that demonstrates thoughtfulness and consideration rather than selfishness. Also, if you're unwilling to do something that someone else wants, rather than outright telling them no, it's better social form to give an excuse cushioned by an apology (e.g. "Ah, sorry, I've already got plans that weekend." instead of "No, I don't want to spend my weekend with you."). Perhaps you already knew all of that, though.
One way or another, learning how to socialize "normally" is simply a matter of figuring out all of those implicit understandings I mentioned about how interactions are typically expected to progress. The best way to do that is to socialize with others more often, as others in the thread have suggested.