A couple things do not make sense here.

Here is your wiring diagram, see the attachment.

The unit is not overfilling so the odds are high that the level switch is switching from VP to VT (the full position).
You can check this by getting a piece of tubing with the same internal diameter as the pressure hose. Attach this to the level switch and start a cycle. If you can stop the water flow by blowing into the tubing then the level switch is probably OK. You should not have to blow very hard.

A bad level switch does not explain why the unit continually gets water while in spin. It simply stops the water flow from over filling it does not route the power to the water valves, the timer does this. Also if the level switch was stuck in the Vt position (which starts the agitation), then you would not get any water as the switch takes the valves and ATC board out of circuit.

My feeling is that you have a bad timer.
Unfortunately the wiring diagram does not give a timing chart so I cannot be absolutely sure.
If we had a timing check you could then use a meter to see what timer contacts are opened/closed during the cycle and which ones are supposed to be..

You are welcome, glad to hear you are up and running.

And thanks for getting back to us. Now when others search for a similar problem they will see what actually worked instead of just suggestions about what could be the cause.