Your girlfriend sounds like she really should get some counseling. She is clearly self-medicating with alcohol when she’s upset rather than handling the issue itself so that it doesn’t keep happening. That’d be the only way she’d get her stuff together… to directly handle all of the issues.

So, yeah, I still think you’re in the right for wanting to end things. She has played a very large role in your feelings by reacting to things the way she does, and it doesn’t sound like there are any signs of that changing.

What you’re planning on saying to her sounds good and honest. If she starts trying to say she’ll change and be better (like most people do when they’re being broken up with), I would stress that you just don’t feel that you’re a match for each other and that’s not something that can be changed. Be strong with it and give her closure so she does not hope for reconciliation in the future… even if reconciliation could be possible if she somehow got her stuff together and her kids and her both started acting better, that’s not what’s happening right now and it’s going to take a lot of time, effort and probably some counseling on her part to change things. She also doesn’t give you that “home” feeling you need. So you don’t want to leave that door open for her because it’s not true of your situations right now, and closure helps people to heal more quickly.  I feel that saying you’re not a match and that you don’t think you’re good for each other will help the decision feel concrete to her.