Thanks to nzzzz, FrizzleFry and, ultimately, Titan_rw for helping me solve this.
I thought I'd post this in case anyone else encounters this problem. It seems that v6 likes to have a trailing backslash on whatever folder is set as NB's data folder in the .nbi file. For me, it was c:\Users\xxxx\AppData\Local\Newsbin in my .nbi. Without a trailing backslash, ie: c:\Users\xxxx\AppData\Local\Newsbin\ none of my Downloaded posts were being marked as downloaded after restarting NB or closing the tab. Adding a backslash to two locations in my .nbi fixed this.
It seems that only DownloadMarker.db3 and Workspace.xml were affected by this problem, and NB worked fine otherwise.