Welcome to the forums! By saying that you are new here, does that mean you are new to Dr. Richards' therapy series as well? I had social anxiety myself, and especially had trouble making eye contact with strangers or even acquaintances, regardless of whether I was talking to them or just walking by them on the street...I would even occasionally blink excessively when talking to someone face to face. The therapy series really helped me see more clearly that eye contact is no big deal and that it is nothing to worry about. I can also look back when my anxiety regarding eye contact was at it's worse and notice that nobody ever said anything about me blinking too much or how I was looking at them, when I was talking to them; maybe they might have thought something about it, but there's no way to ever know for sure, and so there's no reason to worry about it.
The solution from the therapy series for this problem is a paradox...the way to stop worrying about this (and therefore to stop doing it) it is to not think about it. With everything anxiety related, whenever we fight to not act a certain way, we are only making things worse; what you resist, persists. That's why when you think about "blinking unnaturally" or "staring", and try to stop doing it, it only makes it worse and it has a snowball effect.
External focus is a great strategy to apply in these types of situation. Instead of internalizing and thinking about whether you are blinking or staring, try to focus on what the other person is saying to you. If there's too much anxiety to do that at first, you can augment it with doing something physical that will get your mind off the worrying, such as tapping your foot, or something small that will likely not be noticed by the other person and would be enough of a distraction to get your mind off the worrying.
If you keep practicing on focusing externally, you will eventually be able to concentrate more on what people are saying to you, and you will stop noticing what you are doing in the moment. This is the goal of overcoming social anxiety...to get to the point where we don't even think about anxiety anymore. So I encourage you to try this strategy out for yourself :).