More 1957 Chevy Issues 

I heard you say that it would be near stalling unless you opened up to full throttle. That could be a water-in-gas problem. Try gas line anti-freeze. it breaks up the water molecules. If it is water it will start running better within 5-10 minutes. Wish I had a cute assistant.

Had the same/similar issue with mine after getting hot and switching off would not start again unless full throttle. Put a carb spacer on and it fixed my problem

Sounds pretty obvious, gas tanks do need to be vented. Was the gasket around the sending unit drawn in from a vacuum created by the fuel pump