My awkwardness has subsided somewhat by being more confident in my conversations, especially during awkward silence. I used to get anxiety from talking to strangers; not from the conversation itself, but the unknowns (how long will this last? is my story amusing enough for tnhe other person? is it polite to stop talking now? do I need to "wrap up" the conversation?) Now if I don't feel like talking anymore, I'm no longer worrying about whether I'm being rude, I just sit in silence and don't add anything else. Move on to your next thought or back to what you were doing and don't imagine the myriad ways the other person may be perceiving it. Once you don't let awkward silence occupy your mind, you'll find your conversations flow in and out much smoother and you have substantially reduced social anxiety.