He's arranged for his pay to go into his new account, but has agreed to transfer it back across to our joint account to pay for all our direct debits.
"to pay for all our direct debits" - What does this mean OP?
Yes don't annoy him until money has gone in there, for the £250 overdraft to be paid off. That's not the same as trusting him. Then immediately there's no overdraft, contact the bank, explain you've split up and need the account closed or frozen or at the very least for there to be no credit facility attached (ie no overdraft of any description possible and no credit cards attached to the account).
If there's anything you need to pay that's in your name and it's currently coming out of the joint account, contact the company concerned and set up a new direct debit from your new account. If this means transferring some money from the joint account into your new account before closing it, do that.
Why do you both have joint direct debits? Or does he mean your DDs and his DDs? You need to stop sharing an account with him ASAP. If there's genuine joint debts, that you are legally liable for, in joint names (not this additional cardholder thing), then for now at least he's going to have to pay them from his account, because you have no income. He can't insist on maintaining a joint account with you if you don't want one.
Are you currently sharing a house with him, living together?
If you haven't already put in a claim for UC do it right now. It doesn't get backdated to when you split up, only backdated to the date you make the claim and it takes time for your first payment to come through. You can update the bank account information with them when you've closed the joint account. You need money to live on and from this thread you appear to currently have none.
Do you even know what savings accounts he has, how much pension, what his wages are, what debts are in his name, what debts are in your name, any investments in either of your names etc?
If you're not living together, have you removed your name from any utilities bills, rental agreement or council tax? You need to do this. At the moment he's calling all the shots. Don't passively sit back and let him. It's unlikely he's operating in your best interests.
I'm glad you have a new current account. Have you informed the child benefit people to pay it in there? If not, do that now.