My story will not save past the batsman yellow lantern fight. It also won't save when I complete any fights in the other modes. Also when I try and enter the archive it says I need to sign in when I am
Is there space available on your xbox hard drive? You may be encountering save errors if there isn't enough space to save data. Since this is happening with more than one auto-save feature, it might be something you want to double check.
It's also possibly you may have been signed out of your Xbox live account. Make sure you are logged in when playing playing the game.
You will get a 'not signed in' message sometimes if you haven't created and registered a WBID. This is not necessary to auto-save game data, though, only to link your account with the mobile version of the game.
If you are still encountering errors, try to be a little bit more specific about what's happening. I hope these suggestions help.