Frunk/trunk release backlights staying on

[Bobiam]

The 2001 has front and rear trunk release switches by the driver's door. These are backlit so you can find 'em in the dark. The lights are LEDs so they don't take any power to speak of.
My problem is that unless the car is locked/alarmed, the backlights don't go off! If this happens overnight it won't harm the batt. But it killed my 6 yr old batt after being ON for a week or two without the car running.
I just changed the old batt and the problem remains. I can take the cover off and disconnect the LEDs I suppose, but I'd rather fix it right.
Any suggestions? One person already suggested that the ignoition switch may be the culprit. I just don't want to start changing parts all ofthe place until I get some more ideas. Sounds to me that there is a faulty timer relay somewhere.

Thanks......