If you had the stromberg carburetor you could run the motor with the top off and set the float level with the motor running.
Anyway, it seems there are lots of new parts installed and you still have a problem. Here is how I would approach your case:
a) Disconnect the main vacuum line to the brakes and put a solid gasket in place to shut down the power brakes, Bijur, and wipers. If the motor runs smooth then you have a huge vacuum leak. If not leave the system offline.
b) Disconnect the ignition switch at the coil pack and use a jumper wire to battery voltage. If the motor now runs smooth, your ignition cable is shorting out.
c) Disconnect the fuel pump feed and run the motor from a gas can, then run the motor on gravity feed. If you still are having problems you now have isolated the matter to either the carburetor or the ignition system.
d) Check the motor vacuum. A leaking manifold can cause misfires.
e) Check the distributor ground. Packard issued a TSB saying that the distributor tower is subject to poor grounding.
f) If you are still unable to isolate the problem find a fuel injection specialist with an exhaust gas analyzer. If there is lots of oxygen you have an lean or ignition misfire. If you have no oxygen then the motor is running rich and you have a carburetor problem.
Nobody mentioned the accel pump piston. A bad one can work but seep fuel when you don't want it to. Not as crucial on an EE22 and a relativly rare problem, but it's somewhere to look. You said there was fuel outside the carb, so it could be seeping through the vent or coming up the accell pump stem. That's from a lack of needle control. A year ago we had the opposite issue. Our needle/seat assy would stay closed when it warmed up, behaving like vapor lock. It was indeed a mullered up needle/seat assy.