When that problem surfaced on mine, the lamp indicator stayed on "solid" whenever the headlights were switched off, but went off when the headlights were on.
The system is a bit tricky to diagnose, but I'm inclined to believe that the problem lies with the metal rails embedded in the rear taillight housings. Sometimes they get loose over time, and lose contact with ground. So even if the lamps in the socket get juice, the resistance is enough to set the lamp indicator on.
Also, inspect each of the bulbs carefully. If the bulb is turning black, it's on it's way out, even if it still works. That can also change the resistance enough to set off the lamp indicator.
Since yours comes on when the headlight switch is on, concentrate on the bulbs in the rear taillights that come on when the switch is on. Press down on the metal rail near each of those bulb sockets and see if the lamp indicator goes off. If it does, try to crimp the rail down firmly to ensure good contact.
This solved my problem, until the following week when I backed into a truck with a lift kit, shattering the lens...so I just went ahead and bought the whole tailight, so the problem really was fixed!