I hope this message finds you well, present circumstances excluded. I am a licensed attorney with over a dozen years of experience handling matters of this nature. It is a pleasure to assist you and your children today.
You do have options, legally speaking, but you will have to file for divorce in order for those options to kick in, so to speak. If you file for divorce, you can have the court issue a separation order which will detail financial responsibility, child custody arrangements, etc.
In short, he will be forced to help you financially, not only in paying for your lease, but also the bills as well as child support if you are the primary custodial parent...which you most assuredly will be for a number of reasons, not the least of which is that he abandoned you guys.
Once the divorce is complete, you will likely remain the custodial parent and this means he will owe you child support and you will not owe him any alimony. In short, he will have to get a job or he will face other large issues, including potential jail time.
So, unfortunately, in order for this to move your way, legally speaking, you will have to file for divorce and get the separation order put in place. Maybe he will come to his senses once that happens and the divorce will not have to go through. By filing, you get certain rights.
Let me know if you have any other questions or comments. Please rate my answer positively (THREE OR MORE STARS) on the ratings bar on your end.
Thank you and best wishes!