Yes, if you have smoke present, then there's fuel present. You say "cough and choke", that sounds like sputtering to me, an overloaded engine should not cough, choke, sputter or misfire. It should get louder, and only rougher as it gets into the lugging range. So if an engine is rated at 2,800 RPM max speed, then it lugs down to 2,000 it's going to be louder and rougher sounding, and at 1,600 RPM it will shake worse, but it shouldn't sputter anywhere in that range. That sounds like air in the fuel to me, not enough fuel (low pressure) will slow the engine down because each injection stroke is lacking fuel (more or less evenly). Sputtering is one injector gets fuel, and then an air bubble comes through and nothing happens until the air is passed, and it runs again.
Start with the fuel filter, this has nothing to do with oil, probably nothing to do with the air filter. If the fuel filter doesn't change anything, then look at the rest of the fuel system, tank to pump especially.