The three little plastic bushings that go around the gear in the motor have worn out. The motor keeps turning until it catches and then shuts off.
Go to the Portal on the top of the screen and search one of the many retailers there. Most of the major Corvette places like Eckler's, Mid America, Corvette Central, etc sell the replacement little plastic bushings. They are very inexpensive.
It is a little tricky to get the light assembly out and apart to do the swap. I suggest putting wide masking tape around the opening of the headlight hole in the hood and edges of the headlight cover to reduce the chance of scratching.
You should be able to do both lights for under $30 with shipping and a cold drink for your troubles when you finish.
I'll lay odds its your bushings. Go to Home Depot and ask for item number 595609. You will need six of these nylon spacers to repair both headlights. They cost me .69 cents (plus tax) for a pack of three at my local Home Depot store.
Much cheaper than the $7.99 price in MAD, Eckler's and some of the other specialty catalogs and they are the exact same Delrin spacers. Go figure.
If there's a TechTips portal on this Forum search under "headlight bushings." If not, let me know and I'll see if I can get you a copy of the instructions I used when I replaced mine a couple of months ago.
Sounds like it's the bushings inside your headlight motor housings. They grind down to nothing over the years and cause slipping and knocking. They are easily replace, try Mid America or Ecklers, $5.00 for each headlight.