I will be happy to help you out. Can I get the model number to your machine?
If you advance the timer to the spin cycle, will the machine spin?
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I would say you have a bad lid switch on your machine. If the lid switch is bad the machine will not spin or drain. This sounds like the issue your are having. On most kenmores there will be a little tab that hangs off the lid. This tab goes into the ring around the lid through a little hole. The lever that this tab hits to activate the switch should block you from seeing into the hole. If the lever of the lid switch is hanging down to far or has broken off, you will visually be able to see this.
This is a very common problem and a rather easy fix. This is a repair that you could do yourself.
Let me know if you have anymore questions about this. I will be more than happy to help you out further.
It definitely sounds like you just have a bad lid switch.