When the moon comes between the Earth and the Sun, it blocks the sunlight from reaching the Earth's surface and creates a shadow. This is called an eclipse.
Full moons and new moons are just names for different moon phases. A full moon is when the Sun's light is reflected off the entire surface of the moon which gives it a "full" appearance, while a new moon is the exact opposite and the moon will be completely dark. Neither of these cast a shadow onto the Earth.
