I had this problem with a Pontiac grand prix I had. Seems like the most common excuse ppl told me was not enough voltage going to the headlights, so i bought a wire harness adapter and it still did the same thing. I ended up either getting out to make sure both were on or pull up to something and check that way.
I think specifically on the newer (07+?) cobalts the drl's dont provide the proper voltage.
2009 Ford Escape Limited - 3.0L, 6 speed auto, 4x4, 25/33 city/hwy imp mpg.
2017 F150 XLT-XTR - 5.0L, 4x4, 3.55's, E-locker, xtr chrome package, fx4 off-road package.
20/25 imp mpg