When the engine dies, turn the fuel switch to "OFF", then crack the drain on the carb bowl to see if there is fuel in the bowl.
The choke is cutting off AIR when turned to START, which makes the mixture RICHER.
Turning the choke to RUN, makes the mixture LEANER (compared to Choke on START) by allowing more AIR through the carb.
The idle mixture screw (if you have one) will have a head for a FLATHEAD screwdriver. It will be a FUEL screw. FUEL screws meter how much FUEL at idle. Turn the screw Clockwise to reduce FUEL (make the idle mixture LEANER) or turn it Counter-Clockwise to allow more fuel (make the idle mixture RICHER)
It seems like the idle circuit is acting LEAN, meaning too much AIR or not Enough FUEL.
You can try to turn the idle mixture screw to 3 turns out.
Gently turn the screw clockwise until is seats. If it is less than 1 turn to seat it, it is turned in too much. Be careful with it, don't damage the tip of the screw. After it is all the way in, then turn it counter-clockwise to back it out 2-3 complete turns. See if that helps. More turns out will make it RICHER
Also let it warm up a bit before switching to RUN.
And check for leaks around the carb gaskets and adapter. When it is running on choke, spray carb cleaner near where the carb meets the cylinder head. Listen for a change in RPM, or maybe it will die if there are leaks and you spray carb cleaner or starting fluid.
Or maybe a sheared flywheel key? My new favorite solution. LOL
Maybe Timing is OK for rich choke mix, but maybe the ignition shifted too advanced for the leaner RUN mixture.
I don't know. Hope any of this helps.
