... I need to avoid the restore at all costs.
Unfortunately that is not possible.
Once the Device is asking to be Restored with iTunes... it is too late to save anything...
To minimise loss... Connect to iTunes on the computer you Usually Sync with and Restore from the most recent Backup...
Restore from Backup
Here's the troubleshooting article:
You say you need to avoid the restore. Well, you have three choices:
- Restore using a previous backup. I am guessing you have no interest in doing this.
- Restore as a new iPhone. Are you also unwilling to do this one?
- Do not restore, and by not doing so, your iPhone will become an expensive piece of decorative art.
Hope this helps.