Cold water is barely running hot works fine.
You will want to pull the cold water hose off of the water valve to check the screen on the valve. The screen may be clogged with debris which would cause slow fill. If the screen is clear then you want to check to make sure that you are getting good water flow from the hose and if you are then you need to replace the water valve.
Answered by AppliancePartsPros.com | Wednesday, April 24, 2013