If your relays are working properly, you might have a stripped gear in the motor. You can replace the gear or the motor depending on your resources. If the headlight is loose, the gear more than likely has teeth missing.
Haynes repair manual.Chev corvette 68-82($20)
If you go to corvettemagazine.com and click on there tech link they had an article that tells you step by step with pics how to repair the headlight motors
get a new head light motor
You can find a diagram of a 1977 Corvette horn assembly at your local library. Look in the car repair section for a Corvette repair book.
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There are manuals and even forums that can give you information on how to repair your Corvette depending on what your looking for. You can try visiting www.gregsonline.com to order a manual for your car.
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You go to a car repair shop
Have you checked that the headlight switch is working properly?
There are a few shops in CA that offer Corvette auto body repair and training. You will want to contact them for assistance. Try Ken's California Auto Body, Lionhardt Automotive and The Corvette Shop. They should be able to direct you to a starting point if you don't have the experience needed prior to starting Corvette training.