When you click on links to various merchants on this site and make a purchase, this can result in this site earning a commission. Affiliate programs and affiliations include, but are not limited to, the eBay Partner Network.
The culprit is called delrin bushings. There are 3 on each side and are sold as a set (about $7); several of the supporting vendors sell them. In all likihood the gears themselves are fine. The original delrin bushings (rollers) turn into powder after all these years causing the headlights to "flop" around and the motors to spin excessively since they are not properly driving the main gear.
Here is a helpful "Corvette Enthusiast" article that has color photos and step-by-step details on how to replace them.
It really is not that hard to do; take your time and care; while you have things apart, it is also a great time to clean and lubricate parts that need it. I did this on my 1988 Vert and the headlights operate as new again. All for about $14 in parts and as far as maintenance: a bit of white lithium grease for the output shaft and some 3-in-one-oil for the external linkage mechanism.