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I doubt the problem is a relay, there are two common problems with the headlights.
One is due to the stopper under the light housing needing an adjustment, there's a procedure on the forum somewhere for how to properly adjust it that you can probably find once search is working again.
The other, and most common problem, is a damaged gear inside the actuator. If it's a nasty grinding noise that you hear, then that's probably your problem. You can buy a new motor/gear assembly from GM for some major $$$, fix it for free using these instructions, or buy some metal gears and make it last. There are several places that sell the metal gears, do a search on the net for 'corvette headlight gears' and see what comes up.