A couple of things, Ted:

1) First, your car has two two-barrel Zenith carbs, not four barrel. They look big but are not. Unless of couse you have done a marvelously creative carb/manifold swap/fabrication project. Probably not too likely...
2) The fuel return valve on the front carb may be sticking in a dramatic way and allowing the fuel to siphon back toward the tank. It will not allow the carbs to drain since they have needle valves at the floats that regulate the fuel inflow, and an air gap in the carb that breaks any potential siphon. Depending on the condition of the fuel pump valves this siphon may empty the line from the carb to pump or farther on back. Slight chance of this...
3) Most likely cuplrit is the fuel pump itself. There are two valves, inlet and outlet. The inlet valve allows fuel into the pump from the tank when the diaphragm moves up to draw it in, and then it closes. The outlet valve opens when the diaphragm moves the opposite direction to force the fuel to the carb. Both valves need to seal tightly so that fuel cannot leak past. Probably leaking pump valves.

Old pumps could be opened and cleaned, diaphragms and springs replaced and/or the valves could be re-seated with abrasive valve lapping compound, a very short term fix. Newer pumps are sealed with a crimped lid that cannot be serviced, just replaced.

I would replace the fuel pump if it is more than a few years old.

For what it's worth...

230/8