Social anxiety can cause a varied amount of discomfort. Sidkemp has some great suggestions. Relaxing and staying calm are very important, however for someone with high anxiety, concentrating on relaxing, breathing and staying calm can completely distract you from the conversation. I really like the idea of practicing with a friend and slowly moving into more social settings step by step.
Can you identify what it is that causes you so much anxiety in one on one conversations? If you can figure out what your triggers are, then that would be a good place to start. Tackling them one at a time. Look at why they trigger you, where it came from, how can you reduce their impact.
In the short term, when talking to someone face to face, try looking at another part of their face, their forehead, their mouth, their chin. At least it will get your eyes in the general direction, and you can move to the eyes for short periods until your comfort level grows. Also, if you gesture with your hands when you talk, it pull their eyes to your actions, and reduces the need for as much eye contact.
Another suggestion, and this one may seem silly to some, is stand in front of the mirror and look into your own eyes. They are beautiful, and very interesting to examine. Get comfortable with your own eyes first and move on from there. Good luck with your progress, hang in there. Things will get better if you keep on trying. Take care, C.