Kenmore 80 series modej # 110 20842 990

https://www.searspartsdirect.com/manual/3dwsqwqolw-000582/kenmore-11028812790-washer-parts

The are a number of models and they all have numerical model number, '80 Series' is a group based on a theme. It will be one of these.

https://www.google.com/search?q=kenmore+80+series+washer+manual

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387780 does not seem to be a valid model number.
If you can locate the model number enter it here at sears Parts Direct.
Parts for Brands Craftsman GE Kenmore Whirlpool and more

Link to videos about washer not spinning.
top loader kenmore washer won spin YouTube

There is a clear plastic hose that runs from the load selection switch to the side of the washer drum. This hose connects the fill sensor to the load switch. without it being in good condition and connected properly the washer won,t get a signal that it's full.

the lid switch controls the spin cycle, pump probably has clogged

The cubic feet of the Kenmore washer model 110 heavy duty is 2.8 cubic feet. Measured to the top water line 14.5 inches and using the radius of the tub of 10.375 inches.

Hi, It does sound like the inlet valve is bad. If the screen inside the inlet vavle is not plugged, you will most likely need to change the valve. If you will send me a model #. I can tell you how to get to the valve for replacement.

Thanks
Vic

Something in the timer came apart. Timer needs replaced. When in good working order, the timer should ONLY ever be turned clockwise..Reversing this could break it.

Your transmission is not engaging thats why the speed changes your making will sometimes get it to work.Honestly any machine after the 10 year age mark is not worth the cost of replacing the transmission.

Sounds like u may have broken the coupler gear between the motor and transmission... the replaced the belt with these gears on this model... But should go fill & pumb water out and NOT spin Or agitate... IF these gears or broke or stripped... heavy loads like u  said would do this... recheck and see if wash will pumb water out But will NOT agitate or spin.. IF so the above is your problem  and is Not to hard to repair  or much money

Below is a pic with the motor remove the white piece in the middle is 1/2 of the gear system with a rubber piece in the middle (not there)  and another like gear on the motor
sounds like these gears may be broken or stripped on your washer