Awesome thank you for the reply! This is getting a little heavy it but makes total sense, I'm going to think on it for a little while - I don't really want to open the chassis of the furnace, if that's what's necessary to do this, but maybe.
My furnace does use that wiring diagram I posted, 2 for power (for the fan motor I'm sure) and 2 for control (both blue wires). If those "control wires" are connecting the ground, to activate the unit, as you say is most common, then there should be no voltage between them, I assume, but I will check with a meter.
I don't want to mess with the power wires to the unit, obviously, because I'm sure that it's designed to warm up and cool down with "independent" thermostat control, which is why it takes a while to shut off even when you've disconnected the control wires, but your explanation of the solenoid function is great. It's a 2006 Suburban propane furnace if that helps, but believe that they do have several possible wiring diagrams, so I don't know exactly how it's set up.
Thanks again!