I understand that this type of Washer should not Spin until most of the Water is out of the Tub.
Yes, but the motor needs to run for it to pump the water out before it starts to spin.
The motor turns one direction to agitate and another direction to spin/drain.
When the motor first revises direction to drain the water the transmission sits in neutral (neutral drain) until the motor pauses.
During this pause between the drain and spin cycle is when the transmission shifts into spin. After the pause, the transmission shifts into spin and then the motor starts running in the same direction as it was for the drain cycle, but this time the washer spins.
Sounds like you have a bad timer to me. Your model washer uses timer part # 3953321
If you order the timer through the part number link I posted above and it don't fix the problem you can simply return the part to AppliancePartsPros.com
for a refund.