My kitchen sink is backed up in both basins. Plunging just moves water from one basin to the other. Black water and sediment comes up when plunged.

You must plug one side and plunge the other. If that does not work remove the trap, catching 3 cups of dirty water as you remove it. Make sure trap is clear and then rod into the pipe going straight into the wall. You will hit a 90 degree bend about 6" in so use a small 1/4" snake