Is this a Suzuki M120X? I have the impression that it is, but need to confirm that.

We have been blaming your problems on lean mixture, which is probably correct but needs to be verified. Try applying small amounts of choke when the hunting or rough running is happening, and see if that cures it. If it does, the problem is clearly lean mixture. If applying even large amounts of choke does not eliminate the symptom, it will probably not be due to lean mixture and the carburetor is the wrong place to look. The rest of this post only makes sense after you have proved that the engine runs perfectly with the right amount of choke applied. Conversely, if the choke makes no difference to the problem, you will need to look for a mechanical issue in the governor system.

Have you cleaned the fuel filter? Have you carried out the flow test by putting the cup under the carburetor, with the float bowl removed, so you are checking the whole system from the tank through the filter, tap, and hose?

Here is the M120X service manual:
http://golftechs.us/Manuals/Suzuki

As long as you have cleaned the carburetor's fuel intake fitting, so it can't be restricting flow, the only float bowl problems that could be causing lean mixture are low float level, or a sticky or stuck float mechanism. I think the float level is not adjustable, but a problem with a worn needle could affect the level slightly.

To answer your question about the pilot screw, the carburetor's idle system has an idle jet, which Japanese carburetor manufacturers call a pilot jet, and an idle mixture screw, which they call a pilot screw. The amount of fuel flow through the idle system is mainly limited by the pilot jet - the pilot screw is merely an idle mixture fine tuning device. I can usually only make any useful adjustment with pilot screws by attaching a tachometer and maximising the idle speed - there may be no effect on idle quality.

You are aware of the possibility that your problems are due to an air leak through a gasket or some other fitting, into the crankcase or the intake pipe. This can be a tricky fault to find, especially in the case of a leaky crankcase gasket for instance.