When the car is driven for an extended amount of time is when the ets off light comes on. The bucks and carries on when you get on the gas and the light comes on the, The check engine light doesn't go out at all. I thought the ets had to do with the traction control. I am at a lost

With that light on, there is a diagnostic trouble code stored in the ECM.
That code will give you a clue about the rough running/bucking.
ETS is Enhanced Traction System. If the ECM has a DTC stored due to drivability, the ETS will be disabled and the light will inform you of that.
Find out what the DTC is and let us know.

Anytime the CEL comes on, the ETS will be disabled. Using a code reader, find out what code was stored in the ECM.

In many situations, it's a misfire situation. Replacing the spark plugs with the same ACDelco plugs that came in the car often solves the problem. Another common situation is when you keep topping off the gas tank, after the nozzle automatically cuts off, the raw gas damages the EVAP Control System. Check for codes first, however.