Kenmore - Agitate and Spin Problem

Situation: Fills with water, and 4 out of 5 times will not go into wash cycle (to agitate). Instead, motor sounds like it is "humming" but there is no agitate action. After a while, there is a hot/burning smell. This will continue through the full cycle... never agitating nor spinning.

Variable: I can sometimes work the timer (moving it to other parts of the cycle... pushing / pushing out... repeating, etc), and it will sometimes suddenly engage (usually in the spin cycle - and it does drain and spin). And sometimes we can push in the timer to turn it off... wait 5-10 minutes, come back, pull the timer out, and everything works fine.

What I've done: Removed agitator... no problem with dogs and nothing caught inside. Checked drum... nothing caught anywhere. Removed pump... nothing inside. Removed and inspected coupler... no problems. After doing all that, and reading other posts on the site, I figured it was the timer, so I order one from this site. Installed it yesterday and the first two loads worked without problem. Thought I was done. First load today... same problem! And, we pushed in the timer... waited 10-15 minutes... came back and pulled it out... and it works.

So... Not the agitator, not the coupler, and now not the timer.

Any help?

Please.

Many thanks.