I offer you some things I have learned in the last few years. Children do not like their parents being divorced, it makes their lives more messy. They have two parents to visit on holidays, etc. And if one or both parents are angry at each other and involve the AC in that anger, then it makes the AC lives really chaotic and crazy. I am also divorced and my ex does his best to manipulate the AC to not like me, he is allowed to say hateful things about me, and he plays the pity card to them. They should pity him cause mom dumped him and got remarried and he was alone for so long and mom took half his 401K and left him "penniless" and he doesn't have a good job now, yada yada yada. I also have a couple of mean adult children who are narcissists or worse and feel they are entitled to use people and be ugly if they so choose. They are toxic and best to protect myself from them.
If this is your situation, that your adult children feel they have a right to be mean to you, be hateful, ostracize you, make fun of you, etc., then the problem is beyond just your ex and his depression. It is their toxic behavior that will probably never change. So perhaps looking at this separate from your ex, and see if your AC are nice or mean people in general. I always recommend Life Code by Dr Phil to help see the "truth" of people's behavior.
It sounds like your ex has figured out a way to manipulate everyone and make the AC dance to his tune. He gets a lot of mileage from creating a pity party and making you the bad guy. Yes, I hear he is depressed but he is also spreading his misery to everyone -- this sounds manipulative to me. The AC do not like to be triangulated also -- this means do not ask them how their dad is, they don't want to have us asking. Probably is some guilt and manipulation from him that they do not want to convey to you, that they have internalized, or he is still bashing you and they can't deal with it. I know you mean well. It is triangulation when we put them in the middle by saying ugly things about their other parent or by asking questions about their other parent's lives.
Also, our AC do not want to be our friends, our best friends, generally speaking. They see their parents as parents not friends, as separate not tied together forever. We baby boomer generation parents tend to want our children to be our friends and our social life forever. We forget that we were not so tied to our parents as we got older, as we started to create our own families. Also, Dr. Joshua Coleman says previous generations of parents expected their kids to earn the parents love. Current young people generations think the parents have to earn the kids love. So it has flipped, no longer do they give the parent undying devotion and attention just because of the parents role as parent, unless they get something from it and feel they are treated well by the parent. Does this make sense?
Your AC do not see our new partners as anything other than the man or woman mom or dad is married to/dating/whatever. I realize in some families this is different, and there is more respect and love. So I would say do not expect your AC to ever love your new partner, as long as the AC are polite and respectful. They already have a father and he's a handful. You are their mother, BF is just an outsider. This is how THEY look at it. Just possibly how they see it so you will understand the situation better, not being critical of you at all.