If they go down when you start the engine, you know vacuum is getting to the relays and to the actuator diaphragms. So I'm betting it is a bad headlight switch where the valve is not opening when you pull the stem out.
Have you tried the bypass valve under the steering column? You know about that, right? It basically does the same thing as the valve in the headlight switch, but without the lights coming on. Pull the little knob straight down, and the headlights should pop up.
If it works, then it's the headlight switch.
P.S. Welcome to the Corvette Action Center!