Great Video Dick!! Almost like being there! Thanks!
Try roughing up the snubber on the sidewalk. The snubber needs to be rough in order to grip the tub ring so it can limit the movement of the tub. Also make sure there is enough pressure on the snubber bracket.
You should also check the spin bearings for wear.
Put the timer on the spin cycle,with the machine turned off. Hold the spin basket, then turn the machine on. Let the machine on for a few seconds while holding the spin basket, then turn the machine off, while still holding the spin basket still. The machine will now still be in the spin cycle, without the brake on. Grasp the spin basket at two places directly across from each other, then see if you can rock the spin basket back and forth.
A VERY small amount of movement could be considered normal. Howewver, if you get a CLUNK CLUNK with a lot of movement you need spin bearings replaced, and also a replacement spin shaft.
From your video I did not notice excessive cabinet vibration, however you know your Kenmores, so if you say the vibration is excessive, I take your word for it.
Another thing that could be a problem would be the suspention balls. I have rebuilt 6 or 7 old KM/WP machines, and only had to replace the balls on one machine. That machine had sat out in a junkyard.The balls on the junkyard queen were hard as a rock. Usually the balls do not need to be replaced, but your machine might need them. They are available yet, and are not that costly. Or you could try a ball transplant from a newer parts belt drive machine, as they are the same size. If you replace the suspention balls, be sure to replace the lower balls as well as the upper. (6 balls) Also make sure the tub is centered before tighting the balls down. Otherwise the rub will not be centered when the machine is level.
The other probem you are having sound like it might be the timer itself.
Best of luck Dick, and hit me back if you need me.
Rick