Does my ex husband have to continue paying alimony if I am now living with another man?

The short answer is maybe. Usually alimony is terminated upon the remarriage, death, or the entering into of a supportive relationship as defined by the Florida statues, where you're living with another man. The court can analyze the facts surrounding that living arrangement and determine whether those facts taken in total justify a termination of the alimony based on that supportive relationship.