The glowing LED problem is residual power in the circuit. Think about a laptop computer with a transformer in the power cable. When you unplug the computer first, then the wall power, the LED on the transformer stays on. If you then plug the power line back into the computer, the light goes out almost immediately but if you just leave it it'll take forever to lose power. It occurs to me that the glowing LED problem may be just a similar loading of power on the wiring. Yes, it would not be as much power as a transformer/rectifier but is apparently enough to keep an LED glowing.
Anyone who does know basic electronics feel free to correct me on this because while I work on high tech telecom gear I don't ever mess with componentry. I've not had any basic electronics training or any need to use it since 1989
. Anyway, my thinking is that a strong resistor (like 5K - 10K) can be wired from the power wire to ground, attached between the switch and the LED. When power is on, power would take the path of least resistance so would flow through the LED to ground. There would be a very small amount of power flowing through the resistor but since it would be a strong resistor, power flow would be very low. When power is switched off, the LED would now have two paths to ground, one on either side of the LED. Even though the resistor would be very resistive to current flow, it wouldn't have a lot of current to move. This should drain the power out of the LED very quickly when switched off, ending the problem, while not putting the large drain that an in-line resistor would add.
Now I don't have this problem so I'm not going to experiment, but if anyone does want to experiment to see how this would work I would strongly recommend building a circuit to test this, not test this directly on the car. If something didn't work right you don't want to smoke your car's wiring.