How do I cure rough running in my Briggs & Stratton 'Quattro' engine?

After starting the Briggs & Stratton 'Quattro' engine in my Billy Goat KD-10 'Estate' vac, the engine then runs down by itself to low RPM and then it 'hunts' up and down, but never gets back up to max revs. I suspect the fuel/air mixture is 'rich' as it is producing black smoke intermittently. I've already replaced the air filter, and I'm using the correct fuel. The engine starts every time with no hesitation. Does it need a 'carb' adjustment - and if so, how?