I had a similar problem where the air flow would not hold the AFM flap open far enough to get it off of the cutoff switch, so the engine would not idle at low RPM. It ran fine, just wouldn't idle.

If the fuel pump keeps running when you hold the AFM flap open as you described, you should be able to keep the engine running by holding the throttle slightly open after it starts (which should open the AFM flap), or turning up the idle speed with the idle screw (same effect). If this works, then you might have a vacuum leak, which is letting too much air past the AFM flap. Find that or the adjust the fuel pump cutoff switch mechanism.

My problem started when I advanced my timing. The engine needed less air to maintain idle speed so the AFM fuel cutoff switch started killing the pump at idle. I ended up bending the switch mechanism so that it would stay closed at lower air flow.

You can see the cutoff switch and how it works underneath the black cover on the side of the AFM.

I feel this is getting more toward the Fuel Injection section rather than electrical so im going to start a new thread under Fuel injection and quote these last few things.