Is it maybe blinking at a very fast rate, that looks like it's on constantly? Most flashers need a specific downstream resistance to flash at the proper speed, and problems at the end of the circuit such as you're having will affect the dash indicator response. Fix the outside situation, and I'll bet the indicator starts working properly.
Jerry
Edit: there are two separate circuits through a tail/stop lamp. When the ground isn't there, it's not uncommon for the voltage to find a pseudo-ground by backing up to the other side or to a front bulb through the brake light filament and wiring. That will cause all sorts of oddball operating characteristics, such as different bulbs burning with the tail lights on or off, brake application, and turn signal use. I've seen front and rear bulbs "wigwag" with one on, and the other off alternately during turn signal use. That can make the dash indicator stay on all the time.
Jerry
Last edited by Hotrod Lincoln; 08/10/2008 6:55 PM.