Without knowing more about whether or not you successfully authorized your computer to upload from Premiere Elements to Facebook, it's hard to say.
But, meantime, you'll get excellent results if you use Publish & Share/Computer/AVCHD with the preset for YouTube HD to output your video and then manually load that to Facebook.
BTW, how long is your movie? What model of camcorder is your video coming from and what format and resolution is it?
I have authorised Facebook correctly, I don't think it would start the upload process without the correct authorisation, but it completes the upload then shows "unknown error" every time.
The full HD video is uploaded from a Canon 5D mk iii, the resolution is 1920x1080.
I have converted the video to view on computer and then uploaded the file successfully from within Facebook. This is fine but if Premiere has an interface it should work.
I have not tried the YouTube route.
Have you tried using the Upload to Facebook tool with a shorter video -- say 30 or 40 seconds -- just to give it a test drive?
It could be an issue with your internet connection or your computer's configuration that is being complicated by the heavy-duty rendering needed to work with that type of video. (Particularly if this video is in the MOV format, which you haven't advised of yet.)