Dual Mosfet Not Gate Getting Very Hot, Critique Circuit
Remove the N-channel, you have no use for it. They're describing digital logic circuits where you are driving cascades of logic elements and the input impedance/resistance of the next stage is very high. In your case, you have a relatively low impedance of the next stage (tail light) probably being in the 100s of ohms tops. So your output will drop to zero automatically and rapidly without the NMOS at all.

Another potential issue with your circuit is the 3 Mohm. This is going to cause a very slow drop in the gate voltage causing very slow turn-on time for the PMOS. With the redundant NMOS also turning off really slow at the same time, you have a good chance that you've burned both your PMOS and NMOS and that's why you're seeing the 120 ohms across your pulldown rather than the 3 Mohm. Your NMOS is partially shorted now so you definitely want to chuck that part.

Your circuit as it stands would probably work if you had a much lower (say 1-10k pulldown) which would prevent the PMOS and NMOS from turning on too long at the same time. You allowed them to turn on "forever" with the 3 Mohm which causes a direct short from the top of the PMOS to the bottom of the NMOS as they transition from one state to the next.

Summary:
 1. Remove NMOS from circuit.
 2. Decrease your pull down resistance to 1-10k ohm.