In my opinion, i always have hope he gets sober.
It took a long time for me to stop "helping" him. Slowly you do things to empower yourself. You kicked him out, so you obviously are not feeling responsible for him.
The way you do not cave in, is no new contact means no new hurts. The more you see, hear and feel is when you cave. Block the calls, emails, texts. He can find a friend to enable him. Someone will. You don't have to. The farther you stay away the safer you and little jarbs will be. Stay busy and be the best mom you can be!!! You need not let him take your attention away from the kids.
How do you accept this???? Good question. Do any of us accept that the ones we love are addicts??? What I did, was finally realize that I could not longer "help" my addict, I was just hurting him, by enabling him. Give him a home, bed, food, conversation, phone and everything else. We don't have to accept anything, by kicking him out you are not accepting his inappropriate behavior. Right now, accept that you can "love" him from a distance. He doesn't have to be a full time person in your life right now. Take a break, enjoy your kids. Let things happen the way they are suppose to and stop interfering.
Hugs my friend!!! Read, and educate about this horrible disease!!