** The cold water comes in very slowly. ..
There may be sediment screens in either end (or both) of the water fill hose(s) which may be partially plugged. There may also be a sediment screen built into the water valve itself which if it can not be cleaned, the water valve may need to be replaced.
You can read about the water valves used on appliances at the following link:
LINK > http://www.appliance411.com/data.php?info=water+valve
Dan O.
http://www.Appliance411.com/?ref411=Kenmore+washer
The Appliance Information Site
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