The games you've sent invites challenging other people should appear under your current games list.
If is possible, before playing again, can you clear your cache?
If you're not sure how to do this, this link will show you a step by step guide how to do this on your own browser
http://support.chess.com/Troubleshooter/Step/View/2
I'm unable to play on facebook. I cannot find the games I've sent invites for and am unable to move my pieces in the few games I'm able to open.