Washer will not rinse

It seems you have a faulty cold water solenoid

The typical cycle is;
fills to selected water level
agitates wash cycle
drains wash water
spins wash cycle
fills to selected water level on rinse cycle
agitates on rinse cycle
drains rinse water
spins final time
end of cycle

Just read your posting.Now just a couple of questions.Are you using a warm fill for the wash or cold?if are using warm try cold only and take note of the time it takes to fill.I don't think your cold water is filling properly. My manual says that with only the 1st rinse light working it is your control module that is faulty. Please try a cold water wash and reply back so we can rectify this problem.

If your washer is the type that all rinses are in cold water only this would make sense that in the winter months when the water is at it's coldest the clothes will feel cold after the spin cycle.
If your washer has a warm rinse cycle and when you select the warm rinse option the clothes feel cold after the spin, then maybe the hot water supply line is plugged.I recommend disconnecting the hot water hose at the back of the washer and check the screen that is at the fill valve on the washer and make sure it isn't clogged with sediment or calcium build up or any other foreign material that would keep the hot water from entering the washer.If all is well there, suspect either the water temp selector switch or the hot water fill valve. Good luck !

you need to replace the water level switch

Are you on well water? turn off the cold water to the washer and clean the screen filter out. do not leave filter out.
the machine probably has a cold only final rinse so it won't work properly.

When the washer initially fills for the wash, is it only hot water entering the drum? Try switching the water temperature between hot and cold as it fills. If any of them show little or no water getting through, it sounds too easy but check to make sure water supply taps are on, and that the cold/hot fill hoses aren't reversed. Remove the fill hoses and check any of the screen filters for blockage (in the hoses and in the inlet water valve). When the machine stops and waits to fill but all you hear is a hum, it sounds like there is power going to the valve but water is not able to get through it.
Good luck!

most new washers have cold water only rinse and spray rinse ck cold water coming into machine during wash if no cold comes in suspect cold side of fill vale not opening ..could be wiring issue or a bad valve

door switch bad?

POWERUP, YOUR WASHER COULD BE THE LID SWITCH OR THE COLD WATER VALVE. AT THE END OF THE WASH CYCLE, THE MOTOR WILL SPIN OUT THE WATER, BUT ONLY IF THE LID SWITCH IS ACTUATED. IF THE SWITCH IS BAD, IT WILL NOT SPIN OUT THE WATER AND THE TIMER WILL NOT ADVANCE FROM THAT POINT.
IF THE WASHER SPINS, THE NEXT STEP IS TO FILL WITH WATER FOR THE RINSE CYCLE. MOST WASHERS USE COLD WATER FOR RINSES. IF THE COLD WATER VALVE IS RESTRICTED OR FAULTY, THE WASHER WON'T FILL. IF THE WASHER WON'T FILL, THE WATER LEVEL SWITCH WON'T ACTIVATE AND THE TIMER WON'T ADVANCE TO THE NEXT STEP.