That engine is fitted with a manually operated choke via the throttle lever on the upper handle. The choke should 'flick off' once the throttle lever is moved back to the max revs position via that spring to the upper left of the pic. The plastic it is attached to on one end of the spring is the choke butterfly. You should be able to observe that plastic 'lever' flicking off when you move the throttle control off the choke position. If it doesn't move, then the choke is staying on whilst in use and will run rich which will cause the plug to foul.
If not sure it is flicking off when the throttle lever is moved away from the choke position, simply remove the air filter outer cover and the filter itself and observe the choke butterfly through the intake hole inside the filter casing. If the butterfly doesn't move when the choke is taken off, then that is no doubt your issue...however in 25 years on working with Briggs engines like yours, I would be surprised. Yours is an early model hayter and there should a small slotted screw to the front of the carb that will control the fuel richness. It is actually on the front of the carb underneath that pic of the governor assembly. A small long flat bladed screwdriver is what your need to adjust the fuel flow via that screw. Anti clockwise with richen it up, clockwise will lean it off. Do a quarter turn at a time. If someone before has over richened it then you will get a fouled plug, but them may have done that to overcome another issue with the carb such as a hunting issue. Below is a pic of the type of carb, and it is the brass screw you should be looking for.....it's only the early carbs that had this....yours may not have one, in which case it is a fixed jet carb and as you have changed stuff I would suggest already then I would be scratching my head without physically seeing it or hearing it running.