During the wash cycle the agitator moves back and forth to make sure they get clean.
It isn't supposed to spin until the rinse water is to be removed from the clothes by centrifugal force of the spinning drum.
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The gear box drives both the agitater in wash cycle and the drum in spin cycle. It sitsdirectly below the agitator inside the machine. It's an expensive part, and it's difficult to replace. It can only be done by a trained service person, unless you know someone who's done it previously.
It's probably time for a different machine, but it's not a bad I idea to get a phone estimate.
I never buy a new machine. Thrift stores like St. VdeP, or Goodwill have perfectly suitable machines at their stores. I've always had good luck with anything I buy there.