Nope, doesn't matter if it runs in the opposite polarity, the headlight switch could switch positive or negative side of the coil, doesn't matter and making that positive controlled by ignition will fix the problem in either case. But I already outlined above why I'm not doing that.
I thought you couldn't find the relay and that's why you're working at the switch, which is why I figured it is pointless if the polarity is wrong.
the way I see it, none of us (the guys trying to help) have all the information. it is much easier when the car is right in front of you. the statement about having a million of one particular part (though I understand about using what is on hand) may not matter if it can't handle the job at hand.
the drawing that you showed us (hand drawn) showed the part as a TO-92 style package so I decided to look up the 2N3904 (which we used all through first year electronics in college) these things were only rated 100mA for switching. depending on the current that you are running through it, it may never work. once you found out how much current the device will be seeing when the headlights are on, you could parallel several devices together but one device will see the current more than the rest because it will have less resistance than the other just because of manufacturer variations. this will eventually cause that one to fail and eventually cause a cascading failure which all will fail....eventually.
I can't say what is going to work and what is not. I say not to worry about it and to remember to cut your lights off from now on. lesson learned, be more cautious in the future.
looking at it from a cost analysis point of view. I make a descent salary. (overtime is time and a half). I would assume that you have spent hours on this project trying to figure out how to use these parts that are about a nickel a piece. it would have been much more cost effective to get a relay (small, large, whatever) and use it regardless of price. if space was an issue, solder some wires to the splice (srink wrap the connections) and extend the wires to under the dash. some relays actually have a mounting hole in the housing where you can secure it under your dash somewhere with a zip tie. there is the possibility that you are doing this to increase your knowledge of electronics. personally, I feel like that needs to be done on a bread board in a lab ( be it a basement, garage, or even bed room) not on your vehicle which you rely on for transportation.
good luck to you whatever you decide to do. if you do decide to use the transistor, experimentation and data is the key.
Be the change you wish to see in the world
I was thinking about it and don't know why I didn't tell you this before. You can make a relay into an NPN or PNP switch if you want to. It's low-speed high-current and single voltage but works perfect here.
For PNP you have the hot lead in and connect that directly to one side of the coil. Then, you have the load out and the negative lead from the coil that you ground when you want to turn the device on.
For NPN you do the opposite. You have the load's ground come in, the positive lead to activate the switch and the other end of the coil is tied to ground on the load out side.
I used to do this all the time because it eased wiring.
Now you CAN NOT feed positive to the relay coil and expect it to ground through something like an electric fan. It destabilizes the coil and you end up with a rapidly switching relay that will quickly die. For headlights it would work though. Positive from the battery, positive for the coil, and the coil grounds through the headlights.