The problem I'm talking about is detailed here:
http://forums.zynga.com/showthread.php?t=279349&page=167
Essentially, whenever folks click "Share" (such as for thanking neighbors for fertilizing their farm, or for sharing a bonus for a mastery level, or for announcing the completion of building your stable) nothing happens.... the box to Publish the message to neighbors, so that they can claim the bonus, never pops up at all.
It had nothing to do with which Facebook UI you had. I still have the old format, and my wife has had the new format for a few days, and we both have had the same problems, pretty-much at the same times. The problem was probably related to the changing of the Facebook UI, but it didn't matter which UI any specific person had.
There are some indications that this could have been related to DNS, but Zynga hasn't provided any responses whatsoever, which could be used to better understand what has gone wrong. They literally have said nothing about this issue, even though, on their forums, there have been multiple threads about this (not just the one I posted), with literally hundreds, if not thousands of forum members reporting the same thing. (You get what you pay for, and remember: FarmVille is free, if you want it to be.)
Incidentally, the problem seems completely resolved this morning, for myself and my wife.