Need Help With Email? Ask Your Email Questions Online
Hello & Welcome to JustAnswer. May i connect with you remotely to check/fix the issue directly? Thanks
Let's try removing and re-adding account.
Go to Settings -> Mails Contacts & Calendars
Then tap your account and delete it.
Once done, let me know.
If they are IMAP then you will not lose e-mails. You can verify this using a computer by logging into each account and making sure emails are there online via the webmail.