Why won't the spin cycle engage on my washing machine?

I have an older model Kenmore washer that was not agitating/spinning properly, and now seems to be agitating fine but the spin cycle won’t engage.

When there is nothing in the machine, it works fine, but when there is a large load in the machine, it gets “stuck” after draining and the spin cycle won’t start up.

We replaced the following:

• agitator directional cogs (did not help)
• direct drive motor coupling (this had fixed the issue of it not agitating/spinning two years prior but did not seem to make much difference this time)
• clutch assembly (it is now agitating even during a large load but still won’t engage spin cycle)

Thing is that if we’re holding down the lid switch manually and watching the machine, it’ll agitate, drain, and looks like it’s going to start spinning, it makes the noise but doesn’t start spinning. We have to either close the lid and then lift it open and close it a second time or stop holding down the lid switch, give it a second, and then press down on it again for the spin cycle to start. The spin cycle WILL work, but it won’t start on it’s own - it needs to be “prompted”.

Any ideas on what is causing this and how to fix it? Could it be a transmission problem? The transmission seemed to have some oil on it when we replaced the clutch assembly and got a good look at it.

Thank you.

Jennifer