Just a little while after my previous post, I heard from our facilitator. It seems that the office in Ukraine is now closed (holiday maybe?), and the soonest the paperwork can be submitted is July 17.
So, we will have to go back and re-do three documents that will have expired by then. Two of them won't be a big deal, but the other one is a bit trickier, as it is one that has to be apostilled in Washington DC.
And so we wait. I'm so thankful the Lord's timing is perfect.