Yeah, the timer does not decide on the neutral drain, really, all it mainly does is decide what speed and direction the motor runs. Great that you got all those parts though!
One thing though, you mentioned that water comes out if it spin drains. That should not normally happen as I doubt Whirlpool would design a machine to do such a thing. You may want to have a look at your tub ring to make sure the gasket is in good shape and in place. (You may as well, because it is necessary to remove it to replace the inner tub)