Car acts dead after it gets hot...

Dental Dart(all),

Not trying to hi-jack this discussion(lots of good troubleshooting info here), but I think I know EXACTLY what your problem is because I struggled with the EXACT same situation on my 360 duster for about a year before figuring it out. It is your neutral safety switch on the transmission.
For over a year+, I couldn't shut off the car when stopping for gas. If I shut it down, I'd have to sit there for 15 minutes for everything to "cool down" and then car would crank slowly. if I waited another 5-10 minutes, the car started normally. (trying to jump made no difference at all). I am not an expert on how the Safety switch operators on a 727 trans., but mine started to "behave" in this manner after I had my transmission rebuilt. (I expect it is pressure sensitive?)
To by-pass the neutral safety switch out of the starting circuit, you need to run a small jumper wire from the correct lug(I can send a pic tomorrow if you are still working on this) on the starter relay to ground. I haven't replaced the safety switch yet(on my to-do list), but my car starts just fine cold or hot or anywhere in between. Of course, it will also start when still in gear which could cause a problem, but since no-one else drives my car.....
Let me know if this helps.
-Scott