Sorry - should have mentioned in the OP that I'm on a SEBV which is why I'm asking.
If my original understanding is correct, I wouldn't need to go to immigration at all because it does not concern them. Unless I would need to go to immigration simply to transfer my entry stamp from the day I entered on my SEBV? I've seen other posts saying I can just leave the country with my old passport and new passport in hand at the border.