Firstly, if they are dumping their shit on you, it means they are comfortable, which is good, but you need to turn the conversation to positive feelings... and sex
Secondly, what you are experiencing is called "mirroring" and you two I likely doing it to each other. You might start it by acting anxious, which from what you are describing might very well be the case. What you need to do is learn how to relax. Center yourself. So that you are observing rather than mirroring their anxiety and behavior. Meditation is the key to this, but to be honest, that is one of my big problems. Girls get nervous around me a lot, and so they make me nervous. I tend to avoid them after I notice this even though I know that this kind of nervousness is the very best reaction you can get from a girl.
Perhaps trying not to stare people down. It's hard to learn. There seems to be a balance between strong positive eye contact and eye balling people until they become insecure.