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The switch is under the dash on the driver side of the steering column tube. There is a rod that extends down from the dimmer pivot that actuates it. The actual pivot is located in the turn signal bulge on steering column head.
The nut and the screw that hold it in place may be high enough on the column that you may need to drop the column down from the dash to access them.
It feels like the mechanism inside of the steering column is what failed. There's no click when I pull back on the turn signal lever to activate the high beams.
What switch is actually inside of the steering column?
The turn signal switch is inside the steering column head. There is a plastic dimmer pivot inside the bulge on the driver side of the steering column head. The dimmer pivot is connected to the turn signal switch by a small, short arm. It pushes a rod that connects to the dimmer switch when you pull the turn signal lever toward you.
Check to see if the rod going down to the dimmer switch is moving when you pull on the turn signal switch lever. If it is moving, the problem is most likely in the switch. If it doesn't move, then the problem is inside the steering column.
Do you have a standard (non-adjustable) steering column or a T&T? There are fairly complete instructions on how to disassemble and repair either type steering column. They are available at www.corvettefaq.com in the Steering section.