I have an ST3000 with the same size prop. I can get mine to idle pretty low, but had to add an onboard glow and a pump.... not that that has anything to do with your problem, just an FYI.

As far as your problem, I'm assuming your carb can close down all the way? What happens if it does? Will it die?

If the carb closes and the engine won't die, I'd suspect an air leak.