Actually, losing voltage is what you want when you press the clutch switch. The car's ignition needs to read that it's grounded (0 volts). If it sees 12 volts, it won't start.

After the car started up, I decided to unplug the two wires. When I put my voltmeter leads, one on each WITHOUT touching the two wires together, I get 12 volts. If I TOUCH the wires, and then put both voltmeter leads on it, I get 0 volts. So, it looks like all the clutch switch does is complete the circuit, making it 0 volts (very counter-intuitive, you would think that completing the circuit would actually give it 12 volts).

My only other thought is that either the wiring from the ignition switch is shorting out and not reading the correct voltage (ie, it's not reading the 0 volts needed to start the car), or the actual switch is starting to go. I've ordered a new switch that I'm going to keep in the car, and I'll see if the old switch starts acting up again (if in fact that was the problem). I hope my curiosity doesn't leave me stranded somewhere, maybe I should just change out the switch anyhow.

Thanks again!