I would go all the way back to the MPEM and unplug the connection there that feeds the solenoid the start voltage. (yellow/red wire) Test it with your meter there. Between the yellow with red stripe and a good ground. If it is feeding 12V without you pushing the start button, the MPEM is bad. But if it is really doing that, then your solenoid should be switched all the time and the motor would be constantly cranking. So I doubt it really is doing that. Maybe test it with a 12v test light so there is some minor load on it.
Its not 100% constant, its just opposite of what you would expect. The only time the trigger wire is not hot is while pushing the start button.