If you are saying that the machine goes through the cycle as scheduled and spins after the wash but then drains and does not spin after the rinse, then the problem is likely (but not absolutely certainly) the timer. Unfortunately the only good way to determine this is to replace the timer and see if that fixes the problem. Fortunately yours is relatively inexpensive
Replacement parts for MAYTAG MAV4757AWW | AppliancePartsPros.com
so if you are willing to take the chance, I would order one and see if that is the solution.
Good luck, and please post and let us know what you decide and how it turns out.