Hi Camille -
I trust I don't muddy the waters too much, but I have basically the same setup as you do, and two days ago my FB account, got highjacked or whatever. So, having not changed my password in a long while I did that first in FB. I did some other stuff that FB guided me through, including (and this may be where you & I differ) having a 2nd level of FB security, where they - FaceBook - send a code to my iPhone every time i access FB from a different machine (I have six Apple 'puters) OR - from a different access point, i.e. wireless coffee shop etc.
After all this, I was still getting the pop-up as you described. The pop-up took me to System Preferences where I tried to manually input my newly created password. No joy. Today, I went back to Syst Prefs and using the FB Confirmation Code in my iPone Message (seems to be a semi-permanent code). I typed the Confirmation Code into the Password space, and wa-la, no more pop-ups.
I apologize if I've gone to the moon and back, but that's what - eventually - worked for me. Good luck.
Kimball