My iPhone 6 Plus doesn’t run 13, but my new iPad Air does. I’m having terrible problems with Apple Notes syncing from phone, iPad, and MacBook Pro mid 2014. The first advice everybody has for sync problems is to get the latest version of IOS running on all devices. The phone only runs the latest version of 12 and the iPad runs 13. I wonder if that is contributing to sync problems. The web is full of complaints about Notes no longer syncing after the 13 update.
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I’ve had no problems with notes syncing between my Apple devices, but all of mine are not that old except for a MacBook Air from 2011 which syncs fine.
Are you having trouble with the screen on you iPad Air 3? I have to restart once or twice a week, or it starts delaying response. Support seems to has the attitude that that is normal.