You need to replace the water valve, part # 3979346
Here is how it work. If you click on the part number I listed above you will see a photo of the water valve.
On the end that is black there is a thermistor that helps control the temp of the water coming into the washer.
Even when you have a hot water wash selected the water valve is going to pulse cold water to mix with the hot water. Same on a warm water wash, it just pulses the cold water at a different rate.
The washer only uses cold water during the rinse cycle.
All water inlet valve come from the factory with screens in them as JW's photo shows. If your valve no longer has screens in them at some point in time someone took them out.
The screens stop a lot of crap from getting into the water valve. It does not matter if you have soft water or not.
What I often find when screens are missing is that someone else took them out because they were plugged and instead of cleaning them (they are stainless steel screens) and putting them back in they throw them away.
With out the screens there to stop stuff from getting into the water valve the valves plug up and run slow or stop working all together.
Of coarse water valve can fail for other reasons to.
In either case it sounds like you need to replace the water valve.
Sometimes you can find screens in the other end of the fill hoses too, would not hurt to remove both the hot and cold fill hoses to check for screens there too.