I have heard of the same symptom on a 2004 12R - reported by a guy on Badweb.
I went back and forth with a few suggestions and then he stopped responding so I have no idea what his final issue was (or is I suppose).
He started off suspecting a resistive ignition switch - they can definitely cause issues and on a 12X at least there are two circuits supplied via the ignition switch but from your symptoms that doesn't seem to be the issue (unless you had both sets of contacts resistive). You could try 'jiggling' the ignition switch slightly when it is in the ON position and that should show up a dodgy connection.
A buzzing relay is normally an indication of the relay not getting quite enough power to activate. This can be a little misleading because the load that the relay switches in can in turn cause enough voltage drop in a resistive connection somewhere (first step would be the basics - earth and power connection for the battery itself) to cause the relay to drop out. Instantly the voltage will rise because the load dropped off and the relay, which is still be commanded to close, attempts the whole operation over again, and again, and again..........buzz.
You could do basic voltage checks on the keys witch relay - this is the last post for the guy on Badweb:
The low voltage thing could also be caused by a resistive ground on the Key switch relay. It could also be low voltage / poor connection from the battery so measuring what you have would give you a better idea.
When you look at the relay socket the power and ground is provided to the outer two pins in the row of three smaller pins (if that makes sense).
You should be able to measure +12volts on one of these pins (I'm not familiar with your model so I'm not sure of the orientation of the relays sockets) using the battery negative as a reference and the ignition switched ON. The other outer pin will be the ground connection so you need to measure the resistance from this pin to the battery negative terminal. I have no idea of where the grounding points are on a Firebolt but the circuit diagram shows this particular ground going to the same point as the negative lead of the battery. It wouldn't hurt to clean these (and the battery terminals) anyway.
You could also measure the +12 volts with reference to the other outer pin but unless you have a really bad connection the very low current draw of the multimeter won't show up much voltage drop.
If this is all double talk let me know and I can put together a more detailed step by step thing if you like. Do you have access to a multimeter?
I'm afraid I don't have a silver bullet for you but I'm sure we could hunt this down pretty quickly with some elimination.