Check the air hose connected to the water level switch, to see if there are any clogs or restrictions. You can often do this by disconnecting it from the switch and blowing through it. If you can hear air going into the air dome at the bottom of the tub, it is probably clear.

I know you said you replaced the timer, but by your description of the problems, it sure sounds like the timer is bad. Check the connector to make sure it is properly seated.

Let us know what you find.

The switch works by air pressure that builds up as water enters the tub. The part attached to the tub is an air dome. It can become plugged up or clogged with lint or even items of clothing. This will cause a "false reading" of water in the tub, and allow the agitation without water.

Here's a link to a website where you can download a manual for your machine. You may need to remove the spinner (basket) to access and clear the obstruction.

You'll need to register on the site, but it's all free.

ApplianceDigest.com

This is a basic Whirlpool direct drive washer, and your problems are unique, at least for me.

It is likely an electrical problem, and I would check the wiring and try replacing the motor switch. The wiring is the more likely problem, but you know, it could be both.