Does just not going to doctors when sick sound like a good plan for a guy in my sad situation?

Depends. Are you sad emotionally, or have physical illness. If you develop new symptoms, and they are distressful or threatening function, then you should be evaluated. If the sadness is lack of insurance and cash, there are charity clinics and emergency rooms that will treat you until stable. Not being treated always increases the risk of further, acumulated problems.