Gas gauge stuck on full when car is started, moves back to E when car is turned off.
Assuming the tank is in fact NOT full…
I’d say you have an issue with either your fuel sending unit (I think that’s the name) or the fuel pump, or both if they are a single unit.
I know on the late 90s GM pickups, the fuel pump and sending unit were a single piece located in the fuel tank, and the tank had to be dropped to replace it. Hopefully that’s not the case with a 2010 Impala.
This could be the sending unit, but the wiring and Gage cluster are very good possibilities as well. I wonder how many people have dropped a tank and replaced a fuel pump\sending unit only to find out it was just a bad connector. The electrical should be diagnosed first before replacing parts.
Purchase a bottle of AC Delco Fuel System Treatment Plus, and add it to the gas tank at the next fill-up.
Sulfur may be causing the sending unit on the fuel pump to stick, causing your problem.
This stuff removes any sulfur from the sending unit.
Good ideas above, especially about the anti-sulfur product.
If I had that problem myself I’d measure the resistance of the sender at varying levels of fuel tank fill. It appears there’s purple and black/white wires attached to the sender. The resistance between those two wires should vary from 40 ohms to 250 ohms depending on the fuel level. Totally disconnect those wires from the wiring harness before attempting to measure resistance of course.