I don't know for a fact on the Cavalier but on my van it was a separate module. The relay only controls the power to the module and the module supplied the limited current to both lamps DRL lamps. Inside the module I'm suspecting there is a double pole double throw relay . One set of contacts would likely have the reduced current for the DRL and the other contacts would have the full current for the high beams. It's possible the relay inside the module has poor connections and possibly just for the one side or it could be the wiring connector at the module that could be flaky.

 

where is this module located??

 

Just had same problem, tried everything you have. THE FIX.....under the battery tray is a wiring harness coupler....check for corrosion. The battery drips acidic moister onto the coupler and eats the insides away. I had to jump all the wires and now everything works perfect. Hope this helps. I'm in WV myself.

 

Take it to the dealer....it shouldn't cost you but a couple hundred bucks, but then it would be fixed and have a warranty on the repair.