I have been reading through this board with great curiosity. I recently read that a 4.4.4 update is in the works for Note 3. Naturally, Verizon will be the last carrier to push it. It mentioned fixing bugs from 4.4.2, though I have not experienced any unusual issues or anything mentioned in my reading here. Well, not entirely true.... customer service in the field of tech support lacks. Like anything, you have to get lucky by talking to the rare agent who actually has a clue!!
I would love to help if I can. If there is any specifications or settings I can provide from my device for comparisons, please ask.