Ok so he can't do 50/50 because of work but, once he's got his own place, what he can do is have the dc overnight every other weekend and one night in the week or some similar arrangement.
If he should have to move into temporary lodgings and can't have the dc overnight, he can take them out on his days off work until he finds a permanent home of his own.
As the tenancy is in your sole name, he has no legal right to be in your home unless you want him there. Tell him to leave by, say, 8 August and start separating your possessions from his and start packing his belongings in bags/boxes as evidence of your intent.
If it appears he's not doing anything to find another place, make it clear to him that if he hasn't gone by the stated date you'll have the police remove him.
You are wholly and fully responsible for your dc but, as you are not in any way responsible for keeping a roof over the head of their df, you will need to temporarily harden your heart towards him if you want to live a more fulfilling life.
No pain, no gain as they say, but it's likely that any pain you may feel will be replaced by relief as soon as he's gone. Unless he's been violent towards you, there's no reason why you shouldn't stay on cordial terms with him and encourage him to join meetup to make new friends, or invite him to occasional family gatherings etc/round for the odd meal etc providing you don't give him the idea that a reconciliation may be on the cards.
It's only as difficult as you want to make it and, as I can't see that this situation is at all beneficial for your older dc, I suggest you bite the bullet and do it now otherwise all of your lives will continue to be blighted by your unhappiness and his indifference to it.