There are 2 circuits in the carburettor
An idle circuit & a running circuit
Sounds like the idle circuit is fine but the running circuit is clogged.
Not sure of the actual carb you have but on several of them, the shut down solenoid blocks off the main jet & not the idle jet.
This would cause the symptoms that you have mentioned.
Some of the solenoids are polarity sensitive and others are not.
So if they are plugged in backwards they will not work.