You are correct, the differential has nothing to do with the black smoke. Black smoke indicates too rich of a fuel mixture and not enough air. Start with the simple things. Make sure the air filter is clean and there is nothing blocking air from the engine. If all is ok, move to the temp sensor. If it is telling the computer it is colder outside than it actually is, it will over compensate the fuel mixture and provide too much gas to the engine.
Last is the fuel injector system. The pressure regulator could be allowing too much pressure to the injectors. You would need to check fuel pump pressure. Here is a how to video for checking pressure: