What I would do is take a look down the "throat" of that carb as it is running - yes you will have to use you hand on the throttle linkage/cable based upon your description - won't stay running unless you press the gas.

Are you seeing any gas "drops" or anything like that, might need a flashlight to get a good look.

Also look at the drivers/left side of the carb, there is a "sight glass" where you can see the level of gas in the bowl - again with age that glass "window" gets cloudy so a flashlight might be needed.