I've been where you are at now. My exbf (non-alcoholic) cheated on me, left me for the girl he ultimately married. BUT he called me almost everyday and would say he loved me but wanted to be with her...WHAT??? Anyways, i allowed this to continue because i kept taking his calls in hopes that he would miss and want to be back with me after 5 years.
My suggestion would be to stop responding to his calls or emails. He's keeping you in the wings so you don't move on with your own life and get happy. Do what's best for you, cut the ties.