The problem with the final spin could be in the timer.  I do find it slightly odd that there isn't a problem with the drain/spin between the wash and the rinse.  Have you noted whether or not the machine actually drains and spins between the wash and rinse? Also, when you say that it's not draining or spinning is the motor running and nothing happens or is there silence?

Most often when a contact burns, pits or otherwise fails to close in a timer it does so EVERY time that action happens, not just in one spot on the dial.  If this is the case on your timer, the bad contact is most likely the one to the motor reverse start winding.

When it automatically goes into the next cycle does it start to fill or does it dry agitate when the next cycle begins?  If it is dry agitating without filling first you will also want to check the water level switch.  If this switch sticks in the closed position the timer motor will continue advancing at the end of the cycle.

It doesn't sound to me like there is anything wrong with the transmission in this machine, particularly since the transmission is not even used during the drain/spin process!  In a Maytag machine the transmission is used only to operate the agitator.  The advice you got from the appliance guy is very odd as there is absolutely nothing to grease in a Maytag washing machine!  Anything that requires lubrication is oiled with either turbine oil (pump bearing, spin bearing, radial bearing) or a heavy weight motor oil (brake package, interior of the transmission).

Good luck and let us know how things are coming along!

I would certainly go ahead and replace the timer.  It sounds like you have worn or pitted contacts.  The cam may be JUST enough different or worn at that spot on the dial that the motor reversing works in other spots but not in that one.

One trick you will have trying to find a timer for this machine is that it's a suds model and those timers tend to be harder to find since there were far fewer of them made originally.  If you no longer use the suds saver feature you can go ahead and replace the timer with a standard one.  Here are pictures of AND part number of the timers that were used on the A108 machines.  The top two timers were made by Kingston and the bottom one was made by Mallory.  They function the same and the console was designed to accept either timer.

And, as luck would have it, here's a suds timer on ebay for $50 with free shipping!