Kenmore Elite washer will not stop filling

Dirt in your solenoids may be a problem. But other than that it could be bad solenoids in general. It could also very well be that the unit that senses the water level has a part broken. On the inside there is a tube that connects to the tub. When you add water it increases the pressure in the tube and signals the shutoff. Sometimes the connections on this tube go bad.. or the tube itself gets a hole.

Easy way to tell. Start the unit with the on/off switch. Wait a few seconds then turn it off again. If the water flow doesn't stop you're most likely looking at a solenoid problem. If the water flow stops when you turn the switch off then you're most likely looking at a problem with the water level sensor.

Dirt in hot and/or cold solenoids where water enters the machine.