Welcome to the last step of our evolutionary progress: utter self-consciousness. Now that we're gradually approaching the absolute avoidance of the need to establish real-life contact with others, we are about to get lost in the infinite abyss of our own, private minds.
Social anxiety is the mental disorder of our modern times. It's new, and it was born when someone decided to label an ever-felt human emotion—the fear of sharing our mental world with other people.
It has always been there, but as any latent phenomenon, it became alive the very moment someone gave it a name. A vast amount of eyes directed their attention to it, and so it became self-aware. Social anxiety is the disease of our present world.
People have to coexist with people in order to make this society (or any other) work, in order to give it a meaning. We all are, to some extent, socially-anxious individuals. Social anxiety dissipates with practice, and it grows bigger with isolation. In these modern, rushy days, more and more people are attached to their smartphones, personal hand-helds, and tablet computers, secluded from others (if not physically, at least psychologically) through the paradoxical current course of the technology: enhancing mass communications and diminishing personal communications.
So don't feel awkward if you feel awkward while sharing—or at least trying to share—your inner world with others. A time will come when it won't be necessary to share physical space, and so almost all of our communications will happen without the need to look into other people's eyes. I mean, it's getting hard enough to look into our own.