"... if you care, don't let them know, Don't give yourself away ..."
to give away - to reveal or make something known, often unintentionally; also, betray or expose someone
She gave away her true feelings.
He gave away his accomplices.
This idiom is sometimes put as give oneself away, as in If you don't want the family to know about your gambling, don't give yourself away by spending your winnings. (The American Heritage Dictionary of Idioms)