The Daytime Running Lights (DRL) are probably working exactly as designed.
The headlights are automatically designed to be on 'daytime-running-light-low' during the daytime,
without taillights, radio, or HVAC control illumination.
In 'low light' (which, maddeningly, in early Gen 1 cars can be a passing cloud, an underpass, tunnel, or bright daytime sky where direct sun is shaded by the A-pillar, the headlights are designed to come on 'night-time low beam', which also turns on the taillights, radio, and the HVAC illumination. A little yellow-green indicator 'bulb' in the instrument cluster illuminates when this happens.
If you park and pull the hand-brake and turn off the car, then when you
restart, the DRL's are off
until you release the handbrake, day or night. At night, the headlights are on ' 'daytime-running-light-low'' for about 10-15 seconds BEFORE the 'night-time low beam', and then dash HVAC radio, and tail-lights come on.
On the 2003-2004's (and possibly through 2008) it is possible to keep the ' 'daytime-running-light-low'' lights OFF by pulling up one notch on the hand-brake before inserting the key and starting the car. (Useful feature at drive-in movies when you didn't want to turn the headlights on until you exited the parking lot during the movie.)
EDIT: The other thing that happens is that the instrument cluster brightness changes when the 'night-time-low' headlight setting is activated. On 2003's with the 'total blackout' instrument cluster effect, it means that the instrument cluster apparently goes completely black. Beginning with 2004-and later, they made the instrument cluster indicators white instead of red, so they were visible regardless of the DRL function.
Attachment is the applicable pages from the 2003 Owner's manual (the one I have available). There are many threads in the forums about disabling the DRL function, if the 'automatic' features are too bothersome for you.