That's what I've being doing for some time already.
But I want to know, how other people look at learning social skills. Is it sth you can share with others, or will you be labeled as a weirdo or sociopath and avoided by people (which definitely won't help in the case of SA)?
I'm dealing with similar problems, and here is the simple truth I have learned that help me overcome this embarrassment (though not my social anxiety yet, unfortunately):
You would not be so concerned about what other people thought of you, if you knew how little they did.
You occupy the minds of others a lot less than you might think. Random strangers do not care about what you're trying to improve about yourself and how. They will probably not even notice the book you're reading, and if they did, their response would be a mental "huh", in their head. That's it.
Regarding friends and family, it depends on how comfortable you feel sharing such things with them. But all in all, people who will judge you for trying to better yourself are not people worth keeping around.