So you left the kids (abusive) dad 6m ago, and you have got a new relationship with someone you have known for a long time.
Bear in mind that this guy too is going through a divorce, he has been left, and has baggage and issues of his own to deal with. I would say that it's probable that this relationship won't last, as the pair of you have an awful lot of stuff to recover from. As a result of your independent recoveries, you will both regain yourselves.
It may be that the people you are each supposed to be may be compatible, maybe not.
If I were you, take the relationship for what it is at the moment, someone to comfort you, and for you to be there for. Nothing more.
WRT introducing the kids, all being well it would have been a good idea to introduce him AS A FRIEND ONLY, but as your DD is acting up and clearly struggling, you need to focus your attention on her and support her.
Get to your GP and ask for help with DV recovery. If you are in Scotland there is a trial called CEDAR, (Children Experiencing Domestic Abuse Recovery) www.west-dunbarton.gov.uk/community-health-and-care-partnership-chcp/violence-against-women-partnership/cedar-project/ Otherwise call WA and ask for contact with their DV outreach worker for support.
You need to do the Freedom Programme (It's free) it really will help you identify what was abuse and how to recover. It'll also stop you feeling so alone and isolated.
Does that all make sense? The main thing is to be kind to yourself, be kind and patient with your DD and talk to the DC about their feelings. Don't cover up for him, don't take the blame for him, make sure the DC know that his behaviour was unacceptable and for that reason you had to end it.
Again, well done love, you are really doing well.