see what happens when it stops agitating without you touching the washer, tell us of the pauses in the timer and what happens after each pause when it stops agitatimg...or if it just goes dead and sits there till you open and close the lid...
Pegi asked about the pause, did you notice any?
The timer should pause the washer once the washer is done pumping the water out and before it starts the spin cycle.
If there is no pause (the motor stops running) then the tranny can not shift into spin at the end of the drain cycle.
You can create this pause by lifting the lid during the spin cycle too.
If the timer pauses and the washer does not go into spin when the motor starts running again then the tranny will need to be replaced.
How old is the washer?