Washer spin cycle issue

I have a 15 year-old Whirlpool washer. Lately, I've found standing water and wet clothes in it after the cycle has finished.
Further info: - If I let the cycle run normally from the beginning, it fills normally, and the timer dial moves normally. It agitates normally, then keeps on agitating through the rinse & spin cycles, and then stops (without ever spinning) once the dial get to the end of the cycle. If I turn the timer knob and position it back to the spin line and engage it again, it will spin and drain just fine from there. - If I let a cycle run from the beginning, then stop it when the dial gets to rinse (by pressing in the timer knob and then pulling it back out), it will go into spin mode right away and drain properly. - If I let a cycle run from the beginning, then stop it when the dial gets to rinse (by opening the lid and then closing it again), then it will go into spin mode right away and drain properly. - No change in these symptoms in any of the cycles (regular, heavy, delicate, etc)
So, as long as I manually put it into a spin mode, it works fine. But if the cycle just runs normally, then it just keeps agitating and never spins or drains.
Any suggestions or ideas appreciated.
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Sounds like an intermittent lid switch to me. Very common problem. Easy to replace. Fairly inexpensive part.

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