Hmm, moss eh?  You spoze it's grown into the pressure sense tubing?  The fly in that ointment, is that when you advance the knob it immediately goes into rinse-fill.  If it still thought it was full, it would immediately go into agitate.

Two timers do the same thing.  Unfort'ly that doesn't rule out the timer.  More data is needed.  Try setting the timer to the place where it sticks, with the tub completely empty.  Does it still stick?

Meanwhile, verify the sense tube is free of obstruction.

I agree with Tom that the replacement timer is highly unlikely to have the exact same issue as the original, and that the problem lies somewhere else.   We have enough Maytag fans and experts here to help you get to the bottom of this.

This is a real puzzle. I don't have my manuals in front of me since I'm currently at work, but these machines are incredibly simple mechanically speaking, which is one of the reasons they last as long as they do. This shouldn't be that hard to figure out. Once your machine is powered up the tub begins to fill, and as it fills, water begins moving up the inside of the hose that goes to the fill level selector switch. As the water level rises, air trapped in that hose acts against the fill level selector switch diaphragm until the water reaches the selected level, at which point the contacts in the switch are made. At that point the timer will begin to slowly advance and the agitation cycle begins. It continues to agitate until the assigned cam opens the agitation contacts thereby cutting power to the motor. The timer will then continue to advance a little further until it closes the spin cycle contacts and the spin cycle begins. This is why there's a little pause between the two cycles just mentioned. From this point on, without my manuals I'm unclear as to what happens next. I thought the spin cycle would continue until the timer opened the spin cycle contacts, but the fact that it's running endlessly makes me wonder if (as others have suggested) there's some sort of interface between the fill level switch and the timer that stops the timer from advancing to the rinse fill cycle. I'll pull the manuals out and have a look at what's going on as soon as I get home, but by that time someone else may have already found your solution. We'll see.

Does it do this on all three cycles ?,

This washer has cycle buttons for regular, WnW, and delicate, you could have a bad push-button cycle switch.

It does not happen that often,  but we used to see some very strange electrical problems with MT washers and dryers once in a while.

John L.

Your washer fills properly, and once filled to the selected level it agitates, so we should be able to rule out the fill level selector switch since each of those two functions make use of one side of the switch, the two sides being the empty circuit and the full circuit. As you've stated that the problem starts when the machine enters spin(and won't leave), as I see it the only way this can happen is if power to the timer motor is being interrupted somehow. Can you verify that the timer motor stops during spin? If so, in the A407 spin circuit it looks like during the spin cycle the timer motor is powered through a switch between terminals 13 and 15 on the timer. That power also passes through the fabric switch at terminals 13 and 16. If you have an ohm meter you'll want to check for continuity between  those points. The fabric switch seems like a good suspect since the switch at terminal 13 isn't in play during the previous cycles. In other words, the spin cycle is its first opportunity to mess things up. By the way, the orange bar I added to the drawing indicates the period in the cycle when the initial spin is taking place.

Just as a heads up, if you're going to keep this washer long term it'd be worth your time to visit the Automatic Ephemera site and download the various manuals for it. It's an incredible resource and it'd be a shame to not take advantage of it. The link is below and in the upper corner of every page of this site to the left of the sign in tab. Good luck with the machine. Let us know how it turns out.