My ex husband moved out in April leaving me with unpaid rent his name is still on the lease,isn't he still responsible for his share?

Like so many things in law, "it depends."  As to back rent, yes, if both of you were living there, you are both liable for any arrearage, and the court should enforce a claim by you for half the back rent -- if you litigated it in the case, or file a motion for it now.  You say he was your "ex," implying that there is a court order of some sort affecting status -- did the court make any orders as to who is responsible for what going forward?  If not, could you file a motion to amend the order to get a little help with the payments?  As a matter of contract law, the landlord can chase either lease signatory for the rent, but as a practical matter landlords will look to whoever is living somewhere to pay for it.  You may be able to renegotiate, shorten, or cancel the lease if you can't afford to live there any longer.