hard start when warm, then runs rough 1/2 mile

stock 84 Corvette 350
The problem of running very rough actually can happen 2 ways. First after the car has been driven, if you turn off the car it will restart just fine unless you wait more than 15 minutes up to 3-4 hours, then it will barely start and run very rough for about a half mile, killing often, almost killing often, then it is fine. The new symtom appeared last summer on a very hot day 95 deg. Afraid to shut it off because of hard restart I left it idling for 15 minutes. when I then tried to go it did the run very rough for about a half mile, killing often, almost killing often, then it is fine routine. Several service shops have failed to find problem. Any ideas will be gratefully appreciated because this untrustworthy car is now never driven.

I dont know what the problem could be, BUT, my mechanic is extremely familiar with Corvettes ;he has got my 1970 BB running incredibly strong/ rebuilt it from the ground up, he has had at least 4 corvettes in his own family, and has approx. 15 other customers with Corvettes. Even if you are not in the area of Northern Illinois, he would be able to tell you the most likely cause of your problem. Glen Jr. at Glenn and Sons Automotive (1-847-362-2886) . Tell him your a corvette friend of Dave Brown with the '70 Vette---he knows me very very well , and he'll help you out im positive.

Dave

When it is hard to start, press the gas pedal down to do a clear-flood start. If an injector is bleeding down, it could be dripping gas into the intake. See if that helps the hard starting.
As far as running rough, could be a bad plug wire.
The ignition module/pick-up coil could be breaking down when it gets extremely hot, I've seen that happen more than once. Since the fuel filter can stop up quite easily and the 84's fuel pump is low pressure, you might want to change it (the filter) anyway.
Sounds like you might have more than one problem.