Foam Roll the heck out of your lats. I had some shoulder pain (dull acheing) in my right shoulder and none in my left. I had some serious pain while foam rolling my right but none in my left. After I was done, the pain melted away.
You are most likely out of balance. It makes sense that it hurts during pressing. STOP pressing. START pulling and working on stabilizing the scaps and externally rotating the shoulder.