Fuel on the needle valve is likely either "spit" from the carburetor at low-to-mid-throttle positions, or coming from the front of the engine. Some fuel must leak out the front of the crankshaft because that's how it gets lubricated. The needle valve assembly just provides a bit of dead air space so that the fuel can accumulate to a larger drop before it's flung off by the airstream.

If the engine can be set correctly and runs properly, the small amount of fuel can be ignored. It's not likely that it's leaking from the needle assembly, especially at full throttle.

Priming should be done with the throttle fully-opened. You'd block the intake with a finger and prop the engine to draw fuel. The engine should easily draw fuel in this manner.