Confirmation of the problem

Indeed it's the voltage that is the culprit, and there's a solution in sight.

I've created three example circuits;

Top - current and erroneous 12V circuit (should be off)
Middle - current and functional 5V circuit (is off)
Bottom - proposed and functional 12V N-channel circuit. (is on)

Example circuits

( No resistors in these circuits, but they don't affect the outcome here )

Ironically - a while back when transferring the lightboard from design to product, I found that I couldn't use N-Channel MOSFET's, and yet here it is saving me. I must have made some real booboo's on that breadboard when I deemed N-Channels nonfunctional for the lights, since there's really no reason for it not to work - it's all about which direction the current goes.