Why not use copper cleaner.

try lemon or vinegar..

Your lines are probably caused by the soap dribbling off your hands opposed to the water. I know the dispenser for my soap leaves lines on my copper sink. My copper sink came with "Renaissance micro-crystalline wax polish" that is suppose to "freshens colours, imparts soft sheen."

You might have to remove its patina with Mr. Metal and then let it redevelop. I have used the Mr. Metal which makes it look all shiny and then polished it out with the wax. It will eventually redevelop its patina. The renaissance mentioned comes from London, England. I have also used car wax which I seem to use on a lot of the shiny surfaces around the house.