Do I say divorced or single?

There isnt one universal answer. On some forms the distinction may be material for some reason, in which case it is necessary to say divorced. On many other forms it may not be material, and so there is no harm in saying single.

Can I change my Facebook relationship status without anyone knowing?

Hide relationship status change from your friends You can prevent relationship changes from being seen by anyone except you. This means the change wont show up in your friends News Feeds. While editing your profile under Family and Relationships, change the privacy filter to Only Me.