What you're describing is a condition known as a serious misfire. The ECU is actually disabling certain cylinders to prevent damage. In other words, it's running rough because you no longer have a six-cylinder engine. When you shut off the engine and restart, it re-enables those cylinders.

The first thing to do is to check codes. Once you have codes, you can start a methodical approach to diagnosing and repairing the issue. Until then you are just guessing. Some auto parts stores will run codes for you, but be sure you get the actual codes, not just their "diagnosis." Or you can buy a basic little code reader for $16 on Amazon, the MS300, it's a good little unit.

Now, if you want to swap some more parts without actually diagnosing it, a place to check/start would be the air filter (good quality unit like Mann) and be sure the airbox and tube are clear, you may have a nest in there, it happens. Front O2 sensors are also a possibility, as well as the MAF and the intake seal at the TB (from the intake tube).

Hope that helps. Post back with codes after you get them.

Good luck.