Sounds like the "update" may have been bad. Have you thought about doing a complete wipe? Factory reset

 

Try uninstalling any updates to Facebook in Settings | Apps ... then for good measure, click All Apps, find Facebook, click it, and click Clear Data. That will set you back to the built-in Facebook App (with the ICS update, not the Gingerbread one you had originally), and clear login data etc.

Then go to the Play store and update the app. I'm pretty sure they've had an update (not everyone loves the latest updates) since the HTC OTA ICS update. THEN set up your sync again.

I had a nasty problem recently that seems to have been fixed by a factory reset, the first since installing the ICS OTA ... there is anecdotal stories that after any major Android update, factory reset the phone for good measure. I doubted that, but today I'm more of a believer.