Spin basket (drum) does turn in spin cycle

I have a Kenmore 400 (model #: 11029422801) and the washer goes through the wash cycle without problem until it hits the spin cycle. During the rinse cycle, it appeared that the agitator and drum spun in tandem when the water was draining. But in the spin cycle, only the agitator spun and not the drum. I've been advised that this may point to a faulty timer, but a bad clutch seems to make sense too.

I've already replaced the lid switch and that did nothing to address the problem.

Could you advise? I'm trying to decide which repair would solve this issue.