You're human and humans are naturally empathetic to others. You see yourself in other people and can identify with their stress. It just sounds like you are a much more sensitive soul than the average bear, so you're prone to picking up on very subtle cues that someone is not feeling their best.

I guess part of overcoming that overwhelming feeling is learning to trust that others can handle their own problems and allowing yourself to not worry about them as much. Identify that you're seeing or hearing stress in their voice or in their body movements and try to find a way to dismiss it like, they're going to handle it and everything will be fine. You're able to handle your stress, so naturally, they will find a way to handle theirs. If part of the problem is that you feel like you don't handle stress well, then maybe the solution is getting more comfortable with handling stress so that you can trust yourself to handle stress and others to handle their stress as well.