Clothes washer only agitates on Heavy Duty. Other speeds just click.

It sounds like a bad timer, but you need to check it out with a meter. Here are some manuals: https://dl.dropboxusercontent.com/u/8010163/Understanding%20automatic%20washer.pdf
https://dl.dropboxusercontent.com/u/8010163/Electrical%20Theory%20and%20Diagnosis%20of%20Major %20Appliances%20Service%20Manual.pdf
https://dl.dropboxusercontent.com/u/8010163/Electronics%20and%20Electrical%20Troubleshooting%2 0and%20Repair%20Handbook.pdf
https://dl.dropboxusercontent.com/u/8010163/Kenmore_Elite_Calypso%20Washing_Machine.pdf
https://dl.dropboxusercontent.com/u/8010163/Kenmore_Washer_Elite.pdf
Here are some links:https://www.youtube.com/watch?v=fVE9dZODNjk
https://www.youtube.com/watch?v=oBihToN6dxs
You may also want to test the water level switch.

Upon further observation, Agitator goes in one direction only on Heavy Duty setting, back & forth on normal and delicate settings with clicking sound with each change of direction back & forth on hand wash without clicking sound (appears to operate normally) Clicking sound sounds like perhaps a solenoid making and breaking, or two different solenoids, one for each direction.Florescent over head light on same plug as washer, flickers with each click.

With those manuals and a good meter you can find out if you have bad solenoids or, as I suspect, a grounded wire or device. That's easy to find; just unplug it and with one meter lead on ground search around with the other lead until you find the point it reads closest to zero.