Have you tried redoing the update? The latest iOS 9.3.1 update was designed to correct the very problem you're having. Since you're still experiencing the problems, there may have been a problem with the update.
Yesterday, I went to the Apple Store after work and purchased 1 of 2 remaining SE phones in stock. My 9.3.1 issue is now with Apple as I used it in a recycle exchange. I walked away from my i5S with 9.3.1 still having issues. No regrets
When IOS 9 and El Capitan were released, there was an option to upgrade Notes and synch them in iCloud....(it was best to make sure you had El Capitan before you updated because Notes would not synch until you had both systems in place). If I remember rightly, the option to upgrade Notes on my iPad2 appeared in the Notes App, to the left of the title column, i.e. Follow the arrow.