We ran into this in Michigan. They specifically refused to consider a PO Box with the logic that you can't actually live at the post office..
Finally put down my parents address at their hint.
If you have family or a friend, you trust and they are willing, that's a good option for official govt mail. You can always send other things to your PO Box.
If that's not an option, go in and explain your situation and ask what you are supposed to put down. If they don't come up with a good answer, ask them what they use for homeless or illegal aliens when they need an address.