No.
You do not leave your home with two under-5s, when you have done nothing wrong, so the manipulative, whining, porn-addicted but oh so stressed poor ickle manchild doesn't have to actually face things like a grown up for once.
You have MASSIVELY enabled him all through by the sounds of it. You can tell from the way you say there have been 'some issues' and 'trouble with trust' - and then you go on the list a massive sheet of things that would quite frankly be marriage ending for the majority of people. Lying. Alcohol. No sex life, because porn. Phone/internet activity checking, because he's low-level coercive. And I'm guessing that his low-stress job doesn't exactly free him up mentally to take on his fair share of the workload with kids or home.
He's coasted and whined all the way through life and has clearly dropped the threat of suicide/breakdown all through that time to make sure everyone keeps treating the poor little prince whose only problem is he's so unconfident and ^just doesn't belieeeeeve in himself' with kid gloves.
And now, surprise surprise, he's back on drugs.
Your children are tiny. Now would be the absolute best time to get out, before this - him - the dynamic - starts to mess up their childhoods.
He is an unpleasant user, a leech. He WON'T change. He's very carefully helped build up a complete life where he is cosseted because if you don't, it's fallout time.
If you do want to have any hope of saving this sham - I wouldn't - then the absolute only way would be to shock him to the core. You pack his bags, tell him you don't care where he goes, and say if he doesn't you'll go to his workplace and report the drug chat, you'll also ask the GP// SS for advice on having the children around him, etc. Shock him with publicity and make him leave.
You'll absolutely get the suicide phonecalls and the 'I'm here with a bag of pills - any minute now!' shit. You call the police to deal with him and don't engage. He won't do it, this type never does, far to fond of the cushy protected fussed over little life they've always had. And you'll be surprised at how quickly he finds himself a bad and someone to continue the fussing. Family far away? Don't worry, he'll be off to them without a moment's thought leaving you to deal with the kids. Would put money on it.
The ONLY way you will possibly get him actually listening to an ultimatum to divorce if he doesn't pack it in and get a grip is to do this. Anything else will be hot air, because your entire history is you tiptoeing around his pisstaking.
However I can't advise you enough to just get him out and then go straight to divorce and get him out of your life. He will never ever change. Oh and he'll have been hiding his drinking for FAR longer than you imagine - his entire life is lying to you, this will be too.