OneDrive + Small HD + Word = DISASTER

Have you tried https://support.office.com/en-us/article/Learn-about-OneDrive-Files-On-Demand-0E6860D3-D9F3-4971-B321-7092438FB38E? That way you should be able to access any file in OneDrive, whether it's synced or not. For files or folders that should be available when offline, these can be selected with the 'Always keep on this device' option. 

I thought you should be able to open a file from OneDrive or SharePoint whether it's synced or not by browsing the location in https://support.office.com/en-us/article/open-files-from-the-file-menu-97f087d8-3136-4485-8e86-c5b12a8c4176 in Word for example, adding the right location if needed and opening the document directly.

Just to add, make sure your Office is up-to-date, that means versions 1711, 1708 or 1705 depending on how updates are configured and you'll need the Windows 10 Fall Creators Update to try OneDrive Files On-Demand.