Here is an image of the sensor and tube (it's a pipe).
It is just below the idler pulley and kind of behind the tensioner.
Since there are no codes, we have to start assuming it is a mechanical or fuel problem. Have you drained the water from the filter? There is a red or yellow handle (may be black with letters that say drain) by the fuel filter, you turn it to drain some of the water or fuel out of the filter. It may drain onto the ground, so be aware of that.
As for the black smoke, it may or may not under full throttle, so we can't really go by that.
I'm surprised that there are no codes, as the engine running faulty should have triggered a few codes that are unrelated to the problem, but caused by the problem.
I'd replace the air filter, since it made some difference by pulling it. Check ALL of the intake piping and make sure there are no cracks or loose connections. In front of the radiator, there is another radiator looking thing, it is the precharger/charge air cooler. Make certain that it hasn't been damaged or cracked.
You said that the cam sensor was replaced. How long ago was that?
If it was some time ago, it may have gone bad again.
Also, on the driver's side there is a lifting eye bracket, next to it is the Injector control pressure sensor. Round sensor with a connector on the end of it. If you are able to locate it, unplug it and run to check. Let me know if it makes any difference.
My last thought, hail mary pass, is that there may be a problem with the harness under the valve cover. The connector clip sometimes comes loose.