Do you have a soft release attached to the shutter button and/or are you holding the shutter button as you're changing settings, etc.? The LED's should only light up when the shutter is depressed slightly after winding the film. They should then go off after 10 seconds or so as you probably already know. My guess is that the shutter button might be partially stuck down, simulating someone pushing down on it to take a meter reading. If you have a soft release or something attached, unscrew that temporarily and see if the problem continues. Or if that's not an issue, just make sure the shutter button is not being held down by you and if not that, check to see that it's not stuck in a half way depressed position.
On another note, the bulb setting by design turns off the metering lights, so this is not an issue as it's part of the M6.
Hope this helps