This is an easy one.
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.