I assume your using an ipad correct?
Words with friends is all over.......the erros you are getting are related to the app on the ipad, other mobile devices you have words with friends installed and facebook. The first error is related to the account already open on facebook most likely or another mobile device, log out of words with friends on all other devices.
When you load words with friends on the ipad, and your trying to connect to the same words with friends you use on facebook.......then load the words with friends app thru the facebook page from the ipad.
Then when your on facebook and you tap words with friends, it will open the words with friends app and you can play.
FYI - The app does not seem to bring over any "currently" playing games from the facebook page to the app so that may be why you have multiple games going.
The invalid session means you have an open game with this person already (probably on facebook again), so its not allowing you to open another game on the ipad with this person.
The easiest way to solve this issue is to have two accounts........you have the facebook account already so keep that one. On the ipad just use a new name for that account and "do not" load the words with friends thru facebook page if your using this setup. Just get the words with friends from the app store and load it and set it up with a new user id. This will also be easier to follow because if your playing a game and you dont see it, ask the person what name they see.....that will tell you if its your facebook or your ipad words with friends.