Broken Note3 screen with password and debugging mode off need to transfer date any soluations
Custom recoveries have ADB on by default, all you need to do is mount the partitions in recovery and pull data.

My bet is to flash a recovery image via Odin (CWM preferable, because it accepts hardware key control), boot to said recovery (only requires hardware key combo), connect to PC and use ADB pull to pull whatever data out of its misery.