Yes I've experienced this a lot, I have had social anxiety for years.
For me what has helped is exposing myself to the socially anxious situations over and over again, I didn't really do this by choice, I had to, for work each day. Over time the anxiety has reduced, I still avoid eye contact and I still have anxiety but it's more manageable. Have you tried looking just above or below their eyes when talking? Or when you're in that situation focus really intently on a thought or a word while still focusing on the conversation.