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You have a bulb out. Funny behavior...when the bulb goes out, you turn on the headlights and the signal indicator lights up on the dash. Seems messed up but I guess that's how Chevrolet handles the light-out problem when they abandoned the fiber optics in '72.
I've had that exact same thing happen. It doesn't make much sense but it was still a bad bulb. Probably shorting one filament against another. You can check it by swapping your bulbs left to right. If the problem changes sides.....
When the lights are still activated, the arrow is lit on on the dash, but neither is blinking... isn't that the flasher. I don't know if that's the right word, but I mean that silver cylinder that plugs in with the fuse panel under the dash of the driverside. If I'm not mistaken that one is for the single blinkers, but if the hazards are doing it then that part is under the dash of the passenger side. I'm pretty sure that's where they are in my '77. I may have confused you, but I've had the same thing happen to me and that's what I did to fix the problem. They are really cheap at any auto part store or even walmart. Just get the number off the silver "thing."
I had the problem with my left blinker staying on on my 72. It turned out to be the switch in the column. Not the easiest to change and the plug was different on the replacement.
There's an article in the Jan 2002 Corvette fever about light problems being caused by the turn signal switch or harness. I don't have the article but it might help.
anytime I see someone with this problem I always suggest checking for a good connection on this plug under the dash mounted to the steering column. I had headaches with my turn signal(R side) for a week last summer. After trying everything else, turned out to be a loose connection here. Good luck!
It is on mine. Just have to stick your head down by the pedals and look up. Sorry, bad picture, cheap camera. This is where the turn signal switch wires juction with the dash harness.
Finns,
I have exactly the same problem! I haven't had time to trouble shoot it yet so please post what you find to be the answer. (I have checked the bulb).