I have followed your situation closely on this thread, ControversialAnnie - you are getting some good solid advice from everyone and it seems as if you are receptive and realistic (although undoubtedly struggling with the fact you do care about your partner, you have a bond, but you desperately want him to behave differently. It seems he is currently very wide of the mark in terms of his reality-check of the situation, and the fact he needs to step up to being much more proactive and supportive of you.
You referred to the fact that he has not had any role models in his life, which has meant he didn't have anyone to look to for his aspirations (so he basically doesn't have any aspirations now). I'd hazzard a guess that he has only chosen to work because he has convinced himself it's what you want, rather than admitting to himself that it is the only passport to a better life for you, him and your DC's. It hasn't occurred to him that he should be leading the way , not just being prodded with a stick to find a job.
He is telling you one thing, and doing another. He tells you he can't live with you because of the distance to work, and yet when he does have the chance to show you support he is sadly lacking. He is just making weak and lazy excuses and it is letting you and your DC's down. These points have been raised multiple times by posters here on MN, but I have to reinforce it, they are correct!
The last resort will be the "LTB option" - but it isn't necessarily the best approach, not when you have small children. An alternative option might be to challenge him, if he REALLY does want to convince you that he is prepared to change for the better, and not just be "all talk - no action" is to attend couple's counselling together. It might be something you could start off, maybe start with one session on your own. You could begin by off-loading all your frustration, describing everything you have told us on this thread, and get some independent advice as to whether there is any glimmer of hope and some 'next steps'. This would be without your partner listening, so you would have total freedom to "tell it like it is". Then after, when that has helped to sort out the situation in your own mind, you can work with the counsellor to decide how best to involve your partner.
Nothing ventured nothing gained. If it really is a question of LTB, at least you know you have tried your very best to keep your family together - and yes, as people have already observed it currently isn't a cohesive family with him living separately. But you both made those children together, so if nothing else, maybe they ARE the glue holding it all together. You are both young, and maybe your partner will realise he has a lot of growing up to do, now that he does have a family to support.
Perhaps when his behaviour is put under the spot-light by a professional, who know how to handle these discussions objectively and in a way that is constructive not destructive, it will really be driven home to him what he stands to lose and the fact that you do want to find a way through, but not be 'fobbed off' by him with lame excuses and lack of action.
I hope this adds a little to the extensive support you are already receiving on this thread x