@lightspeed2slow What is his character like? I mean, what is he really like underneath it all?
Do you feel like you could trust him alone with your kids for a few hours or a day? Because in a relationship that might be necessary every now and again. (Please, do not "test" him with something like that. Do not use your kids to test him.)
Your kids are important. Their safety, their welfare, their development are all crucial things to consider.
When he entered into a relationship with you, your kids are a part of your life that you can't separate from, so he entered into their lives through you.
From what you wrote about him, I don't think he's the right guy for you.
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Alcoholism is a very difficult thing to deal with. If he truly means to quit, then he needs to get into a support program like AA, but it's got to be his idea.
If he really wants to be in a long-term relationship with you, then he will have to develop some sort of relationship with your kids, or else you will always be pulled back and forth... And that's not fair to any of you, not him, not you, and especially not your kids.
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I was married for 20 years. My wife had a daughter who was 7 when we got married. (Her daughter's birth father was barely involved...he didn't want to be a parent, and it showed.) - I wasn't just her step-dad, I stepped up as her dad. I raised her as my own.
I had three more kids with my wife, and I love all four of them the same... even the one who's not my biological child, but very much the child of my heart.
After the divorce, my ex is with someone else. I had my doubts about him, because he also was a drinker and was upfront with her that he wasn't comfortable around kids, but he made the effort for her.
To be honest, I still love my ex (though she did hurt me pretty bad, because she cheated with the guy before divorcing me). I thought her new man wasn't going to be able to handle giving up his drinking and dealing with the kids. But I've got to admit that he has really changed in those two ways. He did give up drinking (at least the days the kids are there...I don't know about on the weekends when they're with me). Plus, he has developed a relationship with my kids.
I hate to admit it, but he's got good father potential. (He hasn't tried to push me out of the picture. Somehow, he and I have found some respect for each other.)
I wanted to hate him, but I found that he's actually a good man (well, mostly... he still was involved in an affair with my ex). I was hoping he would turn out to be a jerk, but that wasn't the case.
Not just any man can step up and put himself in check, to put the needs of the children who are not his own before his own wants and needs. Guys who can are harder to find... sometimes much harder.
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One more note, and this is just my own thoughts on the matter, but I think it's best to take some time to get to know yourself again after a divorce. I think it's not a good time to quickly get involved with someone else.
My advice is to work on yourself for at least six months before getting into another relationship. There's often some healing and finding your balance that's needed, so that when you do start seeing people you can be more sure of what you want in a relationship.