C4 Corvette gradually dies, 5 min off gets another 50 miles, why?

In reply to parker :

I've never seen a GM module that wouldn't accelerate or maintain speed. They get hot and quit, then cool off and run fine. 

Do you have a vacuum gauge?  If so check for a blocked exhaust.

Update! 

This afternoon I drove around half of Cape Cod doing necessary family things and picking up parts.  No single trip was more than 30 minutes but it was probably 90 minutes of driving total, the car ran fine.  I filled up on gas and left the cap only barely snug.  

The parts I got are a TPS, fuel pump, coil, and ignition module.  When I could finally get to work on the car, it was after dark in a gravel driveway of our house here.  I started with the ignition stuff as that is easiest.   Once I pulled the (nearly new) distributor cap off, I saw a bit of something in the center of the rotor that didn't belong.  Looking deeper, the center button of the distributor had broken in half and the lower half had fallen into the center of the rotor and has clearly been rattling around in there for a while.   Fortunately the only close FLAPS had the distributor cap in stock and didn't close until 9PM, this is one time to thank goodness for common GM parts, the cap was there and $18.  Here is the new center button and the one I removed in pieces:

So, this all makes sense to me.  If the center button was in half, the coil was having to work extra hard to make enough spark to bridge that gap, I can see how it would overheat and have weak ass spark that wouldn't be able to ignite with heavier throttle.  After replacement the car seemed to run smoother.  I leave for the rally at about 4:45 AM tomorrow to pick up my friend and get to the start.  I have the spare fuel pump and TPS in the car still in case they are needed.  

Outstanding news!  I hope that was the cause of the problems.

Another update from the end of day 1 of the rally.  The car ran fine from Cape Cod to Boston and then down to Providence where the clutch master started to E36 M3 the bed.  Limped it along and got to a parts store when it was completely gone and replace it in the parking lot.  
 

Later that day at the end of the day the fuel issue did come back.  Similar conditions to last time include the car running all day and less than a half tank of gas.  Once again a short rest cured it for a while, but this time we tried refilling the gas to fill based on the feedback here that it may be the fuel pump overheating.  That did seem to do the trick and we drove another 45 minutes to the hotel no problem, so seems like that is it, I have a spare pump now to swap in if needed.  
 

Last new issue is the clutch safety switch failed so I finished the day by jamming a random bit of wire over the terminals every time I needed to start it.  All in all an OK day.  

There has long been dialog  that the fuel pump is cooled by being submerged in fuel. An engineer once told me that it is the fuel passing through the pump that cools it moreso than being submerged. I can't say who is right and who is wrong but it does make sense.

I had that exact same distributor cap issue in my XJ. Somehow every time I pulled the cap I didn't look up under it(duhhh) and instead threw sensors and coils and plugs and wires at the car until nothing happened.