"It's always darkest before the dawn", my Mom used to tell me and I've always found that to be true.
I'm a firm believer that nothing happens by circumstance. That everything is planned out by God. The dark trials we go through are the times when we're learning and growing and what doesn't kill us makes us stronger, right?
You (and I, too, in my own situation, but I'm hanging onto Hope) can't see that, yet, but in a year from now..........you may look back and just be amazed at how far you've come.
If you go over to his gf's to get him to sign the divorce papers today, can you take somebody with you? A good friend or an older family member (not the kids). You may be even able to call for assistance at the police department to be there when he's asked to sign.
If he says "No, I won't sign them"....he doesn't have to by law. So, be prepared for that. We can't make them do anything. But, you can take him to court for divorce and get an attorney and IF he contests it then, it's his dime and if he loses, which is likely, then you could get him to pay for your attorney's/court fees, too.
It's gonna be okay. "sometimes God calms the storms and other times, He calms the child".
((hugs))