Yeah you can get him on abandonment. Also I would have his name taken off of anything that you can and try to put a freeze on any joint accounts that you have. Start calling for around for a lawyer to see who can get divorced more quickly and cheaply. You may be able to get into to legal aid and have them do the divorce for free if not at a major discount. He's been gone for 31 days...there's no amount of explaining in the world that would make that a non-abandonment issue. You may also be able to file the divorce yourself. Go into an office supply store such as office max or office depote. They should have form kits for things such as divorce, wills, etc...all of the kits will be in the same place in the store. They come with step by step instructions. Then you place an add in the newspaper that you're going to file for divorce...take the filled out kit, 3 photo copies of the filled out kit, a copy of the receipt that you advertised about the divorce in the personals, a copy of the add posted in the newspaper and go to the court house with these documents...at the clerks office they should give you a court date. Then you go to court, explaine the situation to the judge and it's easy breezy cover girl. When you fill in the kit you will be able to put yourself down for full custody of the kids, what you want him to have for visitation and what you want for child support. The judge may over-ride your request for the amount of child support though...especially if you don't know where he's currently working or where he is. Normally in that event you get what they call reserved child support. Reserved child support means that if you ever find him or where he works then you can take him back to court and have them set up the child support. Words of wisdom...I wouldn't e-mail him or try to call him to let him know when and where court is or that you're filing for divorce....just put it in the personal's section of the local news paper. Nine times out of ten, they don't read the paper, so they don't know about the divorce...believe it or not, it's much easier to get things done more quickly and get what you need if he's not there. If he don't show up they'll just do it without him.
As far as possessions and money and stuff go...you guys are married and everything is half yours regarless who the primary is.