What a complete and utter waste of space. You say your marriage is under strain. I would have thought that is putting it mildly. Doesn't sound like a marriage at all, it sounds like you are supporting a dead weight - a lazy, good for nothing, selfish, twat, who needs an almighty kick up the arse, preferably in the direction of the door.
You've told us that he doesn't care about his kids, he does next to nothing in the house, despite being in it all day while you are out working, he doesn't drive so cannot do any shopping or taxiing kids about, runs up debts on ebay, bleeds the joint account and then says 'tough - as his wife you are liable for his debts'. When he's not sat on his arse in front of the computer buying stuff on ebay, he's sat on his arse watching telly all day. And to cap it all, if you so much as dare to suggest he looks for a job he starts shouting and bawling at you - the one who is keeping a roof over your heads. So what exactly does he bring to this marriage? Without wishing to sound trite he must be bloody amazing in bed ... or something ... that keeps you supporting this pathetic excuse for a husband?
You shouldn't be going anywhere for starters. Can't believe you were considering moving out and paying the mortgage for him to stay there on his own. He's the one who needs to go and sooner rather than later.
Step one is to sort out the finances so he has no access to any of your hard earned money. Get down the bank and get those accounts changed and don't give him new passwords or give him access to the new account details. Once his supply of ready funds dries up, that might change his tune.
Step two is is to to stop facilitating his laziness. By going out to work, paying everything and tolerating a situation whereby he does nothing, contributes nothing, and feels he has a right to get resentful towards you when you complain or suggest he looks for a job you are allowing this situation to continue. Time to put a stop to it. Give him an ultimatum that he either starts pulling his weight with the kids, the house, and he makes serious efforts to get a job - or else. If he shouts and complains, just walk away and tell him it's not up for discussion. Don't engage with him at all verbally. Just tell him you've had enough, and unless he changes his ways it's bye bye.
If you are past that (and I wouldn't blame you if you were) and want to separate/divorce regardless, then go and get some legal advice and start the ball rolling. It might not seem like it, but you are in a very strong position and anyone listening to your story will tell you that immediately. Quite frankly he doesn't have a leg to stand on - morally, financially, any which way.
I honesty don't know how you have put up with this for so long I really don't.