If it only starts with your foot pushing slightly on the accelerator pedal, it could simply be that your IAC (idle air controller) is so dirty that it's not allowing air to get into the engine when your foot is off the accelerator pedal. The IAC is mounted to the side of the throttle body and provides literally all of the engine's air when it is idling and your foot is off the accelerator pedal.
Held on with two small torx bolts, it uses a solenoid actuated plunger to regulate the air it allows into the throttle body from the side. You might get lucky and get it cleaned up enough to fix the problem by using an aerosol throttle body cleaner to clean the inside of the throttle body while the engine runs. Concentrate much of the spray into the long slot you'll see inside the throttle body, on the far side of the opening as you stand at the driver's side fender well. The IAC lets the air in via that slot and the plunger either opens or closes the air hole in the back of that slot based on what the computer tells it to do. To see the slot, you'll need to open the throttle by hand so the butterfly valve on top of the throttle body opens... the slot is underneath that big round plate that opens when you press on the throttle control lever.