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I did not read every line of each post. Forgive me if I am repeating somebody's suggestion. Could not hurt to disassemble and clean the contacts of the bulkhead connectors on firewall.
Try a cheap fix. Find a vette at a wrecking yard (get the same year). Buy the computer and switch it with yours. See if the problem is any different. This will tell you if your computer is bad or not (I had a computer go bad on my 89). If it's not the computer you can sell one of the computers on e-bay and get your money back. Plus this is a switch you can do your self.
I had a similar problem where the car would start but die very shortly after and if I pumped the gas pedal it would stay alive but not be running smoothly at all. I hit a top speed of 35mph getting the car home when I was pumping the gas.
My problem was that one of my computers died. The one in the center console behind the radio. This computer also controls the VATS so it had to be bypassed and then the car would atleast let me use the starter. I had to replace the computer to get it fixed.
I'm not sure this helps but more info is better than not enough. Good luck!
it really sounds like a bad ground to me, funny how a ground can cause so many problems-especially intermittent ones-sometimes the ground makes good connection-but vibration, grease or corrosion will cause all kinds of havoc-check those first.