I figured out another piece of the puzzle - For <340 days, JAL appears to only release award flights for one day in a month for 24 hours, and then once that day is over, availability opens on the next day and closes for the previous day.
So for instance, a few days ago, all the F award flights available were only on Fridays. So Friday, June 18th, Friday, May 21st, Friday, April 23rd.
The next day though, all the F award flights available were on Saturday. So Saturday, June 19th, Saturday, May 22nd, Saturday, April 24th. But now, none of the previous available Friday dates were available. They all say wait listed.
Today, all the F flights available are on Sundays, again shifted over one day. So Sunday, June 20th, Sunday, May 23rd, Sunday, April 25th. And again, none of the previously available Friday or Saturday dates were available.
The switch appears to happen at midnight Tokyo time, so 10AM central. Also it's not that *every* 777 flight has F availability on those open days, but a lot of them do. So right now Sunday dates just opened up, and tomorrow at 10AM CST then you should be able to book F for Monday, June 21st, while any unbooked Sunday dates will disappear until next month.
Hope this helps someone else out too.