Do the lights dim when it spins?
The motor doesn't change direction in agitation mode it just spins clockwise and the transmission causes the back and forth action of the agitator.
In spin mode the motor turn counterclockwise.
If it dims the lights in spin mode as well then I suspect either the capacitor or the motor may be the culprit.

Also don't rule out an electrical issue like a bad neutral on that circuit.
Try plugging in a high amp appliance to verify like a toaster or curling iron in the same outlet as your washer to see if the lights are affected.

It takes more energy to spin the basket of heavy wet clothes than to move the agitator. It might be something in the transmission binding up or perhaps something caught / wedged under the agitator binding it up.