Office 2016 products (Word/Excel) will not stay logged in to O365

Greetings, all.

Today, I've had 2 computers that will not stay logged in to Office products. Here is some background:
We're pushing network shares into SharePoint/O365 Group storage. We're using the most recent version of OneDrive to sync those libraries to the user computers (without downloading the entire library).
When users open a document from the SharePoint library with their locally installed Office product (so far just Excel), It is asking for them to log in again. 
If auto-save is enabled, it asks for the login every time the file is opened. In one case, the login failed. Entered the email address/account and the login window disappeared. Nothing else.
In the second case, the login was accepted, but it doesn't stay logged in when Excel is closed.

In the first case, I disabled Office sync from the OneDrive settings. Now, we see this is creating a local copy on computers with an appended/modified file name. We want to avoid duplicates at all costs. We use these SharePoint spaces to collaborate...not reproduce.
Any tips on how to remedy this?