Is it the bottom that falls out? That happened to my drawer in the coffee bar so I took the drawer to Home Depot and they cut a piece of pressboard to fit and I glued it in place and now no more sagging or fallen bottom to the drawer. Maybe what you store inside the drawer is to heavy for the bottom to hold.
Here you go!
It sounds like you need drawer stops. One of the links above addresses that.
Are you saying the whole chest of drawers falls forward when you open one drawer fully? OR The bottom falls out of the drawer when opened?
It falls forward