OK - I need to think about this a little more. You've tried all the usual fixes.
One last thing - try un-installing an app and then re-install it. To delete an app, hold down any icon on the Home screen until they all start to jiggle. Then press the 'X' in the top left hand corner. A message will appear asking you to confirm that you want to delete this app and all of its data. Then go to the App Store and re-install the app and see if the problem persists.
You don't need to pay to re-install apps, but - in any case - try it with a free app, if you have any, or one you don't care too much about.
If you don't have any like that, then try installing a new free app - anything - and see if the problem persists.
Tim