Engine starts sometimes, but won't stay running?

I have a 1983 Nissan 200sx that I'm trying to get running again. It starts and runs strong, sometimes, but most often not. The first time it ran, it was started by pouring fuel on the air filter. It ran strong, idled fine, until, I assume, that fuel ran out. I replaced the spark plugs, wires, fuel pump, fuel pressure regulator, fuel filter, and some vaccuum hoses. I also replaced the aged gasoline in the tank. It started, in a plume of white smoke that quickly went away, and ran fine until I shut it down and for a week. Now, if it starts, it runs rich. I put a test light to the injector harness and all four injectors are a receiving a signal. When it is started, you must feather the throttle to maintain idle. If you press the throttle more than 1/4, it will stall. Starting the engine and letting it idle on its own will eventually stall it, though it will run rough during that time. Feathering the throttle, you can get the rpms up to redline pretty easy and hold them there, though, again, using more than 1/4 throttle stalls the engine. Once the engine is above 2000 rpms, it runs smooth, but is still subject to stalling. It is easy to push start. If too much throttle is given, even at higher rpms, the engine makes the same sound as if you were climbing a hill in too high a gear. If fuel is added to the fuel filter (or poured in the air intake box), it will run strong and have no problems until that fuel runs out. Sparks also exit the tailpipe as well as soot. Oddly, there are times it will run fine, and after observation, it doesn't seem connected to temperature or moisture. It is easiest to start when jumped, but if it starts, it will start so long as the engine cranks. Even if it won't run, it will start within a couple turns of the starter, then die. Currently, it does not smoke at all while running, but you can smell the excess gasoline from the tailpipe. The spark plugs also seem to foul with soot. I have replaced those twice. Much thanks for any advice!