The washer works fine except for the following. The tub will not spin. It won't initiate the spin cycle by itself. If I spin the tub by hand, then the washer will complete the spin cycle successfully. The timer works fine and advances through all cycles including spin.

I think that the shifter is bad.

If it is a belt drive - I would say one of 2 this is bad either the wig wag (that is the real name) or more likely your pump pulley is going bad and sticking.
If it is a direct drive - Sounds like the transmission is on the outs or you spin tube is dragging. If once it starts spinning it spins slow it's the transmission.