I too am having issues, I just upgraded to Office 2016 and running Word version of 15.12.3.
I can not open ANY existing .doc or docx files, it always says I have to grant the file access, and I follow along and nothing works, I can't even save new written documents, says there aren't the correct permissions. This is very annoying!
I can't believe I paid for this upgrade and have lost a crucial part of my workflow!
The image below is me trying to open a random file I had a coworker open on their computer (worked fine) and send it to me and it does not work.

This photo is me writing a new word document and trying to save it on my desktop, no dice.
I hope that someone can shed some light on as to what is taking place, but so far I am very upset at this upgrade, thanks!