I had a problem with my light staying on when the car was running.
The guy at the auto electrical shop said it was an easy fix just something inside the alternator
needed switched. It would cost like 5 bucks.
But with it staying on even with the key out is really strange. do you have a glove box , trunk or console light that may be causing a power draw.
You can check for a draw with a normal test light. Clamp end on hot side hold the light probe to the ground side terminal keeping the light tight on terminal remove terminal from the Batt.
If light goes out you are good if it stays on you have a dead short someplace.
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Patrick
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*SOLD * 1971 LeMans street car, 428ci TH350 3:31 12 bolt, 3880#,13.6 @ 101.82 Show & go 
1964 Tempest work in progress  400 with Tremec 5 speed
1967 LeMans drop top project. I tracked down my great grand father & dad's old car.
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