Quote from OP...(((I love him. I just hate the liar that he has become. I can't live with it. I deserve more. I want to be married to the "old" him.)))
Who is it you love? Who is it you hate and can't live with anymore? Who is it you want to be married too?
Honey, I fear the "old" him you were once married to and loved so much, is NOW the alcoholic, liar and maybe cheater you hate and can't live with anymore.
Hauling this new "man" into mandatory rehab will almost certainly just be a costly and futile exercise for both of you.
As for his lies and contact with this other woman, I do so feel for you....but only you can decide if being treated with the disrespect of being lied to or possibly cheated on is something you can bear to live with.
You deserve so much better for you and your children, but unless you actively decide this and begin doing all YOU can to get that "much better" for you...it is not going to happen. Face it, your man is too busy satisfying HIS needs and wants to care much for anyone elses, because that is a requirement of alcoholism...it demands all their time, effort and loyalty.
I wish you all the best for what lays ahead of you.