That doesn't work. Has nothing to do with the problem. When the screen is locked and you hit the home button the unlock screen appears for only a few seconds and goes black ....sometimes before the keyboard appears and almost always before you can type in your password. The solution you offer is only of use if the phone is unlocked and it's timing out after inactivity
I am also having the exact same issue as the OP. Very frustrating when you can't even type one character of the unlock code. Sometimes after unlocking and I swipe down to clear some notifications, the phone goes black and locks. Sometimes it unlocks and lets me start to do something (i.e. Internet, gallery, camera, etc.) and wham, it goes black and locks before whatever apps finishes loading. Sometimes it locks after the app loads. And sometimes it doesn't.
This is almost as bad as the "Unfortunately, Contacts has Stopped" every single time I open the Phone App. I have a entry for that issue, so don't address that one here. Please go read it in the Android Forum. Titled "Phone closes with Contacts Stopped eror" Hmmmm, Looks like I need spelling lessons.
As I looked through the other list of complaints/issues in this "Note 3" forum, I saw where someone made a suggestion to Wipe the Device Cache. I followed the instructions and my phone app issue went away. Maybe this will also correct the locking screen issue I have experienced randomly.
I went most of today without having the lock screen blank out before I could start typing the pass code. Then for about 5 minutes this evening, I again had the problem fight me. Clearing the Cache Partition did solve my problem with not being able to make phone calls.