I had this problem too. I read many posts about this issue, tried various workarounds, and found a method that worked for me. Maybe you can try it:
- Completely delete the Facebook and Messenger apps.
- Backup your device (which doesn't have Facebook and Messenger).
- Restore your device to factory settings.
- Go through the welcome screen without signing in with your Apple ID.
- Once you reach the homepage, use a different Apple ID to sign in to the App Store and download the Facebook and Messenger apps.
- Log in to your Facebook account. (This worked for me.)
- After that, you can sign in with your real Apple ID and iCloud.
Before attempting the above method, I recommend that you simply delete these apps, log in with a different Apple ID, re-download the apps, and try logging in to your Facebook account first. I didn't try this because the above method worked for me, but I think logging in with another Apple ID without resetting to factory settings might work too. It's not too hard to try it first.