when a 4-stroke motor with a carb is running lean, because the carb is fouled and not letting enough gas in, then if you lightly and slowly close the choke if the engine stabilizes and tends to have more power then that validates a lean condition. And in such a case the spark plug(s) will be clean.
You stated your spark plug is black, and also when closing the choke the engine dies. That is a rich condition. And assuming you were intelligent with closing of the choke and not carelessly flicking the choke fully closed not knowing any better, knowing you gave it only 1/4 choke valve closing and 1/2 at most and the engine dies then it is definitely running rich. most motors like this you can run full choke and it will run just bla bla bla and not smooth.
carburetor 101 - a higher fuel level in the bowl will always result in a richer mixture, and a lower fuel level in the bowl will result in a leaner mixture. Setting float height to achieve the correct fuel level in the carb is always step 1 as everything else is influenced by that. Carbs like this, you usually can't set it, they are not made to be purposely adjustable, and to do so requires bending parts that were not meant to be bent, and if not careful you can make things worse so be careful when deciding to do so.
and your mention of shutting off fuel and it seems to run better just before running out of fuel... also indicates a rich condition is happening until it leans out as it runs out of fuel and fuel level in the bowl drops to the sweet spot level where it runs right for however many seconds.
on a 1-cyl engine with carb like this, without a fuel pump, the only way a rich fuel condition can happen is the float is not floating and is allowing for higher fuel level, or the needle is not seating allowing fuel to gravity feed in making for a higher fuel level in the bowl. The orifice in the jet(s) don't get bigger and magically let more fuel in.
but first thing I would do since this is used equipment is replace the spark plug for $2 with an new ngk of the correct heat range. And then if it still runs rough and that plug turns black, remove the carb float and check on the bench in a cup of gas that it floats and then make sure the needle is clean and then when assembled with carb in your hand you blow by mouth into the fuel inlet nipple and hear air enter the carb. Then you turn carb upside down - so the float closes the needle against the seat - you should not be able to blow air into the carb by mouth. if u can then the needle not seating is the problem, and you can do this check immediately after taking the carb off the motor. my guess is dirt + age the needle tip is worn/dirty not seating letting extra fuel in. The float if white plastic rarely ever go bad on their own it's because someone cracked it or bent its mounting piece to where it results in a too high fuel level in the bowl.
