I have a strong habit of putting up a front around other people.
It's cold, formal, to the point, and "get me the eff out of this interaction as fast as possible."
It's also very habitual because I usually only realize what I've done after the fact.
A few things have been helpful.
Mindfulness of the body for one. People tend to carry tenseness and aversion in certain parts of the body. For me, I suck in my gut and my breathing gets a lot shallower. Maybe you cross your arms. Maybe your neck gets tight. Relaxing these specific tensions helps a lot, in and of itself.
Remembering that this person will also get sick, grow old, and die. What's more, they likely have not come to terms with any of it and will suffer greatly when it happens. That can get yourself out of a "me and my anxiety" mindset into a more relational one. You have this suffering human condition in common.
I read something on I think the Zen International forum about how meeting someone is meeting a totally unique and dynamic manifestation of the entire universe. Everything from the physical bodies to the mental climate to the circumstances of the meeting are results of myriad mostly unknown conditions. It's a cause for wonder and curiosity, if you can remember to look closely enough.
I'm not sure if any of that is helpful, but it's gotten me to at least treat strangers more graciously, and not always go into interactions with a GITTIR DONE approach