Okay, I found a "typical" GM DRL circuit diagram (Haynes) and have a couple of things for you to check:
1. With the headlights OFF, does the DRL indicator light on the instrument panel? When you turn the headlights on, does the DRL indicator go out?
If so, the module, headlight switch, and DRL relay are working. If the DRL relay were not switching, and everything else was okay, only one head light would come on. Don't ask, crazy circuit design.
2. If you turn the headlights to ON and switch between high and low beams, does the high beam indicator on the instrument panel go on/off? If not, the problem is likely in the dimmer switch, not too uncommon on GM with a lot of miles and/or age. If the switch has broken internally, a common failure mode would be that it won't pass voltage to either the low or the high beams, nor would it send the high beam signal to the DRL module which outputs the signal to light the high beam indicator on the instrument panel.
To try and simplify, the headlight switch on the dash tells the DRL module if the lights are on or off, and sends power to the dimmer switch in the steering column. The dimmer switch then passes the power to the high or low beams, and also sends a signal to the DRL module telling it the high beams are on. The DRL module then turns on the high beam indicator in the instrument panel.
Just to be clear, the PARK lamps on the front do turn on and off with the dash headlight switch? If they do, that indicates the circuit breaker for the headlights is passing current as it feeds both the headlights and the park light circuit.
Let me know what the two tests above do, and you should have your answer. There is always the possibility that you have an open wire in the steering column, but unless some one has worked on the column, not as likely the issue.