That's a good thing, There's a good chance there is no short in the wiring. Remember when you unplug the switch, you unplug the power for the choke light. Keep in mind no new part is good until it has proven it's self ! I just gave a quick check on my 84 and the wiring is like this, Power from the choke fuse brown/white goes to the switch, from the switch there are two wires out light blue to the choke and dark blue to the choke light.
With the engine off and the ignition switch in run you should have no power at the choke or the lamp. Unplug the oil switch and you should have power on the brown/white.
If you have power on the brown/white install a jumper on the oil switch connector. You should have power at the choke and lamp.
If the lamp goes out the oil switch is bad. The switch may not close all the way, it may show 12 volts coming out, but it cannot let enough power thru to feed the choke. What little is getting thru can power the lamp because the lamp uses little power. But good news the circuit works and the lamp is telling you there is a problem! Don't remove the jumper for the oil switch yet.
If the lamp is on and you have 12 volts to the choke, the choke is bad or the choke ground is bad.
You can take a wire and ground the choke, just go choke ground or carb mounting stud to clean bare metal part on engine. Did the light go out? fix the ground. The choke relies on a ground path from the intake thru the carb mounting hardware to the carb body. Plastic spacer? Hell, did you spring for the plastic carb body? Did the plastic carb come with a extra ground tail for the choke?
You can do one more check to see if the light goes out, you need to put a bigger load on the choke wire. The easy thing without spending alot of money may be to buy a 1156 lamp and lamp holder, if you wire it to the choke wire and touch the lamp holder to ground, the 1156 lamp should go on and the choke light should go out.
If the choke light does not go out you may have a problem with "bulb test" in the ignition switch.
!!!!!!!!!!!! jgarrett "bulb test" does not apply to you, You stated that "the choke light goes out when you unplug the oil switch" if your choke light stays on with the oil switch off then bulb test would apply to the problem!!!
I did not realize this until I was finished writing it, I am only leaving it for reference for other people with choke problems.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Put everything back the way you found it, remove the jumper ect.
The ignition switch has a bulb test feature in "start or crank". You may have noticed the dash lights up like a Christmas tree when you go to start the engine, this is the "bulb test" feature. There two extra contacts in the ignition switch, one that switches positive and the other switches ground. It's there to check the dash lamps. The one that switches positive may be stuck closed and sending power to the check engine light.
If you look at the top column just above where the column mounts, looking form the right side you will see ignition switch. Bunch of heavy red and pink wires, at the left side of the connector there are two small wires. One is "black" switched ground, the other "tan" switched hot (+positive).
With the ignition switch in "run" and the oil switch unplugged (no jumper!) check for power at the tan wire. You should have no power. If you have power the switch is bad.
If your feeling brave ( I hate cutting up factory wiring, but it will save you the trouble of changing the switch) follow the tan wire down 3 inches or more from the ignition switch and cut it. Leave enough so it's easy to re-splice. If the choke light goes out, change the ignition switch. If you want tape off both side of the tan wire untill you change the switch.