Did this problem start with the new carburetor?
Sounds like the float is set too high and now allowing the needle to open. Depressing the "tickler" presses down on the float, sinking it, and opens the needle valve. If it only runs after pressing the tickler it means the everything else is good.
Check your float level. How is your carburetor mounted? Is it tilted one way or the other either vertically or horizontally? Is so, how much? Got a photo of it?
Tom