Sorry its turned out to be a long one!
I know this sounds really harsh and its easier said than done, but leaving him and not considering what he may do to himself is your best option. As others have said, he is not your responsibility. You've said he's attempted suicide before but also are aware it wasn't enough to actually kill him. Yes there is a chance he genuinly wanted it to work, yes there's a chance any threats of suicide are also genuine. But given the info you have shared, there is also a very high chance he is taking the piss, using mental health/suicide threats as a way to guilt you into staying. If he genuinly feels suicidal he would have likely acted on it by now, conveniently though, he's only coming out with these feelings and threats when you have had enough.
Once you stand your ground and don't react or respond to any threat he makes or any actions he carries out, he will get the message and stop. He's doing it because he knows that ls your weak point.
Call social and tell them what you want to do, tell them about his threats and your worries. They will help you, or at the very least they will put someone in contact with you from an organisation who can help. It's hard leaving someone and evan harder with kids involved and an ex who makes things hard. There will be someone to help you navigate your way out safely and they will be able to help manage some of the admin load when it comes to housing. He is emotionally abusive at the very least and social already have this on record from their previous involvement. Things like that will back up your housing application so it will be good for you to get them involved and have their records for evidence with the council.
Are the kids on contract phone or pay as you go? If contract, contact the providers and get their numbers changed ASAP and do not give them to him. If pay as you go, just get them a new sim card with a new number. That solves the issue of him contacting the kids with his guilt trips. If they are pay as you go, keep hold of the sims and check them occasionally for threatening messages off him, keep them if he does send any. They could come In useful in the future if things get nasty.
Get the number for your local crisis team. Once you've left, if he contacts you claiming to feel suicidal just send him the number for the crisis team and tell him to call them. If he contacts outright threatening suicide, call the police for a welfare check and let him know you've done this. If he contacts sayings he's already done something, (overdose for example) call an ambulance and let him no.
If you keep doing these things rather than engaging with his threats, he will get the message that you no longer see him as your responsibility and he will stop with the threats if he's not genuine.
Should his threats turn out to actually be genuine then you will have contacted the correct service in Any scenario and they will provide him the help and further support, not you.
You are absolutely not hopeless. You are just trying to take on to much responsibility and consider people who are not your problem. He's a grown man who can sort himself out, but it would be easier for him to play victim and be full of self pity so that you then sort his life out for him. Don't fall for it. You've got yourself and your kids to sort out, he isn't thinking of ways to make it easier on you and the kids so stop trying to think of ways to make his life easier. I'm sure there are ways you could get him out the house but I understand you saying you'd rather just go yourself. So think of that as you doing your bit to help him out. You've made sure he's still got a house while you and the kids have fuck all and start from scratch. That alone is more than enough so please stop thinking you owe him more to make his life easier and yours harder in the process. He certainly isn't thinking of you and the kids in the same way.
It's good the bills are in his name. Is it rented direct from the landlord or through an agent?? Before you evan leave the house you need to make sure the tenancy is dealt with.
The last thing you want to be dealing with is rent arrears racked up in your name, for a home you don't Evan live in. That will screw you over massively with several things, it has to be a priority before you leave.
How old are the kids? My message is already long enough so I won't add much more 😅 they won't resent you though. I've been in your shoes, as have many others. I know there's a lot of guilt there for the kids in this situation. Once you're out and settled, you will be kicking yourself for getting so worked up over the guilt and resentment worry, and you will quickly realise that the kids were only ever suffering in the old setup living altogether, not the new set up you will have.