What should the questioner do with her neighbor who harms her with lewd singing and refuses to stop, and what is the ruling on moving from the house—is it cowardice?

If the departure from a harmful neighbor is for the purpose of getting rid of their harm, then that is not cowardice but a correct action. It has been said: "A dwelling is not set right until the neighbor is set right," and "By its neighbors, dwellings become valuable or cheap."