The patient is a 1996 Chevrolet Caprice with dual-filament bulbs. Both filaments in both bulbs light up in high beam mode, but no filament on either bulb lights up in low beam mode. What could be broke and how do you fix it?
The bulb modules could actually be bad, despite both filaments coming on in high beam mode. Pull the connector at each bulb, there will be three wires: tan, green and black. The black is ground, tan is low and green is high. Make sure there is power on the tan wire when the headlights are on and the high/low switch is set to low. If the tan wire is hot, replace the bulb modules. The bulbs will be easier to get out if you remove the battery and the air box. Each bulb is secured by a plastic collar. Rotate it clockwise to disengage, then pull the bulb out and undo the connector. Reverse the process to reinstall.
If the tan wire is not hot, look for the dimmer switch on the side of the steering column. The connector may be accessed without taking off the bits of the dash or dropping the column, though removing the access panels will make it easier. The outer plastic panel/hatch comes off after removing two screws. Removing that will reveal a metal panel with four bolts. Removing that opens a space to access the column wiring, though you may need to remove the metal collar around the column. The connector has a yellow wire, a tan wire and two green wires. Jump the connector for the yellow wire (from the headlamp on/off switch) to the connector for the tan wire (to the bulbs). If the low beams come on when bypassing the switch, replace the switch. To get at the screws/bolts holding the switch to the column, you might need to remove the lower dash and drop the column. The lower dash comes out if you remove all the screws (you have to take the ashtray out to get at one), open the glovebox and gently pull. The column is secured by two bolts. Don't forget to undo the shift indicator cable before dropping the column. It just clips on to the shift collar (note where).
If the low beams don't come on when jumping around the switch, check the wiring between the switch and the bulbs. The tan wire from the switch goes into the body harness and through the firewall. There is a connector (C101) around the washer fluid tank that would be a convenient place to check if you can find it (look for the green and tan wires going to the bulb on driver's side).
The most likely scenario is bad bulbs. The next is a bad dimmer switch followed by wiring faults at various points.
It's probably something as easy as changing the solid state relay in the fuse compartment in the engine. You can swap the fan relay with the headlight relay to see if your headlights come back on. If so, go buy another SOLID state relay for replacement. You don't want to drive without an operating fan or headlights. Very rarely do both bulbs go out at the same time as they are independent of each other. Try this fix first, not that rocket science above.