I thought I'd add an update to this thread.
The lubrication of the mechanical timer ultimately did not resolve the issue. I gave up and purchased a new unit and had it installed. I figured that after 18 years, the old unit had passed it's useful lifespan and time to invest in a new unit. The new unit is working great. After install I did have an electrical issue where the circuit breaker for the washer kep triggering. I resolved this by replacing the circuit breaker.