Hey! Been fiddling with this. I have 2 exchange accounts that kept disappearing moments after adding them, and only my Exchange accounts (iCloud, Gmail, non-specific IMAP all remained). Tried restarting, deleting the V6 folders in the user Library, and reinstalling macOS Mojave. None worked. I then turned to Keychain Access after reading some success elsewhere on the Apple forums. Here is what I did and it has worked since. (Attaching screenshot of Keychain Access as well)
- Open Keychain Access (spotlight search is fastest)
- On the left-hand side, chose All Items
- In the Keychain Access search bar in the upper right, search for exchange
- In the results below, select all items that have com.apple.account.Exchange.oath-### and Delete them
- Quit Keychain Access and go Preferences --> Internet Accounts and re-add the Exchange accounts
This has worked for me, and all 100,000+ emails have re-synced without the account vanishing. Hope it works for you!