This is 7 years old. I removed the bottom service cover and looked in. I immediately saw a broken shock absorber. I saw there was an unbroken shock absorber on the right side.

I ordered a pair of these. We are advised to replace both at the same time. What looked like an easy job was not. The nylon pins that held these shock absorbers in place did not come out easy. They have little keeper tabs on the end and must have a 1/2 socket placed over them in order to get them out.
:mad:
I got out the broken one on the left. I was unable to replace the pins with access from the front service panel. I had to remove the back panel. I was then able to drive the nylon pins in with a screwdriver and ball peen hammer.

The shock absorber on the left was a bitch. First the water pump and associated rubber drain plumbing had to be removed. Then I had to removed the control board at the side of the washer. This control board absolutely prevents the removal of the nylon pin holding the shock absorber in place.
:mad:

I finally got the shock absorbers replaced and tested the washer. It worked fine. I replaced the front service cover. This washer is in my basement against a wall. The rear cover is not very easy to remove. I left it off. I realized that the belt on the back had never been replaced. I will order that belt and replace it. I have no pets or children to worry about. I will leave this rear cover off in perpetuity. It will make repairs easier.

I am simply posting this to help anyone else having this trouble with this Kenmore washer.
:)