In venacular terms, it sounds like your unit just needs a tuneup but there may be more to it. What is the compression of your engine? Can you see the valves moving freely? If it is spitting fuel out of the intake it is getting too much fuel. So why then is it not going into the cylinder and burning efficiently?
Since your spark plug is black, dry, and sooty, you obviously have a great deal of carbon in the cylinder. In this case, it would be advisable to remove the head and clean the head and pistion top. Carbon may be holding a valve open or be built up on the valve stem causing it to stick from time to time.
Another thing I would do prior to this is put an inline spark tester on your spark plug and watch it while the unit runs. Maybe your Ignition Module is overheating and cutting out on you and that is why you have so much carbon and the unit is stalling out no matter what you do. Cleaning the fuel system and setting up your carb properly is definately a good idea but it may not be the solution. I have seen many times where even an experienced technician chases a fuel problem and doesn't notice that it is the Ignition Module at fault.
Checking the fuel cap was a great idea but since you now know that isn't the problem continue searching for other possible issues. We know the unit will run so now we need to know why it stops even with the choke on. Leave the fuel system for now and check the spark. I would definately put a new plug in it but since that didn't solve your problem (are you sure you didn't get a bum plug? I happens.) look to the module and see if that solves your issues.