Headlights won't work 2002 silverado 1500 hd

9/10 times if you have no low beams it is the multi-fictional switch on your steering column

Check the bulbs could be blown low beam

Light switch faulty looks like,

There is a headlight relay in the engine compartment fuse box, but its after the headlight switch and before the dimmer switch, so it powers both high and low beam circuits.
I would check the dimmer switch.

These trucks are known for bad multifuction switches.
the switch that i talk of is the signal switch , the handle is the switch, it bolts to side of steering column . this switch controls the signals and the headlights and high /low beams. the other thing is the headlight switch connector, the connector some times goes bad and starts to melt or just comes out of the connector. you will need to remove the cover to steering column, the screws are located under the bottom cover in holes. remove cover and check connector and then test for power if you have power going into switch but not coming out then switch is bad.

the last thing is to make sure the bulbs are good , make sure the headlight bulbs are good , and also the driving light bulbs, test for power at the connectors to those lights to.
You will need to do a little testing to find the problem.

I would start at the headlight switch check your ground wires and connections at the light themselves also check the bright dim switch

The wiring for the low beams would run not to the ignition, but to the dimmer switch. I don't know what color they are, but sounds like you are at the source of the problem, if the low beams quit when the column is let down. Pull the trim pieces off the steering column and examine the wires going to the dimmer switch. You may see some wiring insulation scraped off. It could be a bad dimmer switch, but check the wiring first.
Good luck.

This could be a faulty light switch at the steering column

Sounds like the plastic tab that your High/low beam lever (on the steering column) switches has slipped out or broken and was displaced by the column movement. The lever assembly will have to be pulled, inspected, and (probably) replaced.

Considering all you have done, that is the only thing left (except a broken or shorted wire that connects to the suspect lever assembly - but unlikely).
Let us know what you find... please! -Jim

Headlight switch on dash???