If my husband is going unaccompanied to Korea while I move the family back home, what's the best way to get the military to pay for the move?

The military will only pay for one PCS for this, not to include your husbands. That being said if you do not have a follow on assignment and you get reimbursed for a DITY move then when you are traveling to your next assignment they are not required to reimburse you. They only cover one move for this instance and that is his PCS to Korea, the second move is intended for moving from where you are and where he is to the next assignment. That's what my wife and I did when I was non-vol'd to Korea. Since my household goods were with the wife I could claim the travel from her location where she was collecting BAH and then I still got reimbursed from my travel from Korea. Hope this makes sense, and helped you in time.

Pay for everything, keep all receipts and mail them to him in Korea. Resolve passports issue and the EFMP situation. Sounds like this will be a busy time for You.