Kenmore TL Washer Just Hums

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

You need what is called a motor coupling. can be found here. Very easy to replace.
http://www.repairclinic.com/Washer-Dryer-Combo-Parts?s=t-thin+twin-%3d%3da17

Try this link it should help

replacing a lid switch

This assumes that you have a basic Kenmore / Whirlpool top load, direct drive, neutral drain (it doesn't spin until the pump removes most of the water) machine.

First, see the Sears parts site for your washer:
http://www.searspartsdirect.com/partsdirect/index.action

Enter your model number (###.######) and you'll see a list of major sub-components with diagrams and parts.

Since it does everything else, the problem points to the timer.

See the following Whirlpool / Kenmore installation and service manual for top load, direct drive washers with an extensive troubleshooting section.

https://secured.whirlpool.com/Service/SrvTechAdm.nsf/2cd44500d572193285256a45004fd9d6/f7e2147fa98b28cf85257122007475cb/$FILE/L-55.pdf

Your complete model number including the first 3 numbers , is needed to help you get to the coupling . The coupling , is the piece ( pt # 285753 ) which connects the motor to the transmission . When this piece breaks , the motor will run , but no machine action , except for water drain . Even spin , will not work , yet , you will hear the motor running .

Bad/broken coupler/ http://www.davesrepair.com/DIYhelp/DIY285753.htm

Recommend contacting Kenmore directly for assistance Kenmore Customer Care Customer Service Information

Make sure all springs are still in their place tilt the machine back and look under it and you can see them. 2nd all legs must be level as well as the floor.

Unless your front loader is set firmly on concrete and leveled to perfection, the force of a heavy wash load spinning in excess of 1000rpm will create a bit of vibration and noise.A firm footing is your best defense against to much movement.
Also due to the earlier movements during the spin cycle, the shocks that are used to dampen the spin cycle might have been broken or damaged.
If the problem persists inspite of proper levelling, please do check for the shocks or might have them replaced them for good.

where is the lint trap located