Yes, I am sure the switch was part of the problem. A new switch, 59.00 at AutoZone, which surprised me, fixed the "won't turn off" problem. Now I have low, high, off, and washer. No delay. It looks like the O'Reilly guys may have sold me the wrong pump/circuit board some time ago, or the whole motor assembly is bad. I'm not 100% sure if the delay is controlled in the motor relay inside the assembly, or in the pump circuit board on top of the assembly.
They now work, they wash, they park correctly, I'm happy for now...