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Do you have a model number so I can better assist you? And when the washer doesn't agitate, do you hear the motor running or is the unit 100% silent?
OK so unfortunately this means that your transmission has failed. No the agitator coupler should not be able to slide up while bolted. It sounds like your transmission shaft isn't properly geared into the transmission anymore. You'd need to order and replace part number WH38X10002 if you wanted to resolve the issue, and that is a lot of work. If you had a technician come out and replace the part for you, they will charge about $350. I'm very sorry about the bad news.
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Hello again. I'm just following up because I haven't heard back from you and I see this question is still open. Is there anything else I can help you with?