The moon does revolve around the sun. It revolves around the Earth (slowly falling towards Earth on a long slow downward spiral) at the same time as the Earth revolves around the SUn (slowly falling towards the Sun on a long slow downward spiral).
Actually, the moon is moving farther from Earth... by about 3.8 centimeters per year. The Earth is moving away from the Sun, if you could call it that...about one micrometer (a millionth of a meter, or a ten thousandth of a centimeter).