To forswear is to give up an idea, belief, or habit that you’ve had previously. New Year’s is a popular time to forswear anything from sweets to bad relationships.
When you forswear, you abandon something completely. In Romeo and Juliet, Romeo is at first smitten by the maiden Rosaline, but once he lays eyes on Juliet, Rosaline is history. He says of Juliet, “Did my heart love till now? Forswear it, sight, for I ne’er saw true beauty till this night.”