As said before, clean those intake vent screens, front and side, looks like it may have collected and run down with the side panel paint removed... The washer hose has been leaking for quite a while from the severity of rust, have you check it during a wash cycle? Maybe the water on concrete source..... Use the Stainless steel, no-burst water supply, as mentioned.
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17,000 dryer fires a year, when did you last clean the inside of the dryer near motor or the exhaust ducting?
I'm unfamiliar with that particular unit. However, my furnace humidifier has a control the same as the one pictured. The same instructions based on the outside temperatures.
On my humidifier there's a 3/8" copper water line supplying the water.
There's a water trough inside with a float that allows the water to be turned off and on.
Sometimes calcium thats in the water fouls the valve that controls the water.
If this occurs the trough over-flows and water runs out onto the floor.
There's calcium in the over-flow water and when the water evaporates the calcium (salts?) is left as a powder, that looks similar to flour.
From the photos that you have posted, I would suspect that the internal mechanism is fouled by the same crud that is apparent on the lower screen.
The unit should be taken apart, cleaned and any worn or damaged parts replaced.
As for your washer hoses, these should be turned off immediately as they appear to be on the verge of bursting.
I had one of these hoses burst and experienced major flood damage.
There are hoses available that have a metal jacket and I would recommend you would buy this type.