If I come across an individual who doesn’t identify as their birth gender or any gender (so they ask to be referred to as they/them) what is the proper way to go about this? Do I still refer to them as he or she or do I call them based off of what they want? And how would I maintain the best mannerisms while navigating this?

Do what is best in protecting yourself from harm. If by calling them by they/them is supporting their claim then it should not be done. But if it’s a norm in society and not deemed support for this behaviour then it’s ok