Car cut off when stopped on 2006 Chevrolet Cobalt

My car shuts off when i stop at a light, or sign, etc. When i start it back up, the gas needle drops, and slowly goes back up. But this all occur when my gas is under half a tank. When it's full, there is no problem at all. Also my rpm needle goes up and down, and my car stutters. What is causing this?

Asked by for the 2006 Chevrolet Cobalt
possible fuel filter, fuel injectors, or water might be inside the tank