Not sure what a "bronze adjuster" is. I'm only familiar with the BFD models.Those carbs only have two adjustments: idle fuel circuit and a linkage screw for setting final idle rpm. The idle fuel screws on them have a limiter cap on it so that it cannot be tampered with. You actually have to break it to alter the adjustment from the factory setting. Not sure about yours although I suspect that it is similar except it may not have the limiter cap. I will caution you that cleaning these carbs can be a bit more involved than most people think and if you do disassemble one, you should have, at the very least, replacement "O" rings and seals for reassembly. There is one oring that is critical to the proper operation of the carb and when it is disturbed, it should not be reused.
Running the engine "dry" is a fairly accepted practice but if you live in a hot climate, I don't recommend doing that. Burning a lean mixture creates high combustion chamber temperatures and could lead to the fuel "detonating". This usually results in pre-ignition of the fuel and causes the engine to backfire through the intake which causes valve and carburetor damage. Running the engine until it stalls also does not completely accomplish what you are trying to achieve anyway, which is to drain the fuel from the carburetor bowl. Honda has provided, and now recommends that you use, a drain screw in the float bowl. It is attached to a rubber hose and drains the float chamber COMPLETELY out the hose and away from the engine with a port located just next to the water pee port. Only this method will ensure that any and all sediments in the bottom of the fuel chamber are expelled. Again, your engine may not have this, I don't know.
It is admirable that you try and drain the carb after each run as that does help keep trouble away.