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Actually, that would cause air of the wrong temperature to come out of the outlets. A malfunctioning A/C controller or programmer are what would likely cause the OP's problem.
When I worked on mine, the programmer needed new solder on a couple of connections (work done by another corvette owner) and the vacuum source for the programmer was disconnecte and needed a vacuum check valve. Once the programmer was soldered and the check valve was added, all of the HVAC doors worked fine and still do after 2 years.
I can't swear this applies to early cars, but L98's and LT1's both have a vacuum check valve in the supply line from the intake manifold. The black and white plastic piece shown in photo here.
It is common for the ports to become brittle and break off or the valve itself to fail eliminating vacuum to the AC controls. Typically found on the intake manifold near the vacuum supply or in the vacuum line near the firewall. Most parts stores carry the valve for less than $10.