Yes, lots of ideas, no easy fix. First question for you is does your vehicle have Daytime Running lights, and are you confusing DRL for headlights, they are the same bulb, only dimmed.
Yeah, Toyota's idea of daytime running lights is to just leave the headlights on. Supposedly it's dimmer than when you turn them full on, but I couldn't tell the difference.
There's a wire you can cut under the dash that will disable it. Or you can get fancy and wire a switch into it so you can turn them on or off.
Yea after further review I believe they are the daytime running lights that are staying on. There is very little difference in the brightness between the running lights and headlights.
Just to be redundant, yes, with an '02 you've got drl. You also probably need to put new bulbs in because said drl tends to wear the bulbs to the point where they're dim as hell.