Where to start...
1.) Is marriage counseling an option? That is, do you WANT a divorce? Even if you don't, your STBX can file. AZ is no-fault.
2.) Does the base have any legal services available? Checking with the Marines website for Yuma, I found something about "LifeLines," which may give some general info:
(Click on "relationships" in left menu, then "divorce" on the next page.)
Check with ANY family counseling services on the base. They must have something... Even if they don't provide any legal services, they probably know where you can get some advice/assistance. The Yuma County Community Legal Aid Office can be contacted at 928-782-7511.
I don't think "the state" will help you find a place, let alone pay for it. Further, once you are divorced, neither of you may be entititled to live in base housing, I think. Not sure. You can file temporary court orders to ensure monetary support from him. Filing a divorce petition needs to be done in order to file temporary orders, because you need a case number. Being in the military, I would think getting his wage attached would be pretty straight forward. If you do get temporary orders in place, you may have some financial support from your STBX that may allow you to seek other living arrangements, if necessary. The medical coverage for all of you will have to continue until at least the divorce is final, by law. After that, I don't think the marines will continue to cover you medically--but maybe will the kids--after the divorce. Also not sure. Refer to LifeLines website above regarding the 20/20/20 rule. How long have you been married? Probably not 20 years... You mentioned going on maternity leave, which imples you have a job. What resources does that provide? Medical benefits, income, free legal services, etc.?
3.) Get an attorney if you can. Since your husband won't cooperate, I don't see how you can do a divorce by yourself.
4.) Either way, here are some websites with information and all the forms and instructions, for Mariopa & Pima counties. I cannot locate any for Yuma county as yet--there's nothing on the Yuma Co., website. READ THE ONES BELOW anyway. They're most likely VERY similar if not the same forms you would be using.
http://www.cosc.co.pima.az.us/home.asp?include=pages/divorces.htm (GENERAL INFO--Pima)
http://www.superiorcourt.maricopa.gov/ssc/forms/fc_group_4.asp (DIVORCE WITH CHILDREN--Maricopa)
http://www.superiorcourt.maricopa.gov/ssc/forms/fc_drtmc1.asp (TEMP ORDERS--Maricopa)
By reading these forms over, you can get a clearer picture what needs to be done. You need a crash course in the ins-n-outs of how to get divorced. Doing this will help you decide whether you need an attorney or not. If your STBX doesn't get one AND you become sufficiently "educated" about the process, you may be able to do it yourself. However, with three kids, I would personally be afraid of not securing some legal advice.
For Yuma county forms, try your nearest library, office supply store, etc. Also, I believe there are some other considerations/forms given that it will be a "military divorce." This forum has a section on military divorce. Check it out:
http://www.divorcesource.com/info/military/military.shtml (Probably mostly general info...)
5.) It costs $272 to file a divorce petition in Maricopa county and $242 in Pima county. It's probably less in Yuma county. Theoretically, the original filing fee is the only money you'd have to pay, but there are always other costs, such as copying documents, etc. If you go with an attorney, it will cost a bare minimum of $2500. It's typically more like $3000-10000. Document preparation services which advertise "a divorce for $300" or some such amount are
a.) Not lawyers; they're paralegals--at best, and can offer no legal advice
b.) Charge you for completing the paperwork--you still have to pay the filing fees
c.) Helpful for those who cannot/will not learn how to complete forms themselves, but don't want to/can't pay for a lawyer.
Finally, I hope this gives you something to chew on. Sorry I couldn't be more specifically helpful vis-a-vis Yuma county. Their website stinks.