Then the only thing left I can think of to try would be a full "Hard Reset". Unless you wish to take it to a tech and have it looked at. However, they may well do the same thing. Make sure you back up all personal data from the phone to a cloud account or a computer. Especially since your Google access is not working correctly. You can't rely on the phone having synced to anything to save the files, music, videos, contacts, etc. So be sure you back it all up first.
NOTE: Doing this will clear 'all' the data on your phone, so it is suggested you not choose this option unless you have to, or aren't worried about losing anything on the phone.
Be sure you back up any data you want to keep, to a computer or cloud account, before doing the...