Thanks for sharing the logs. When was the last that you restarted Fusion? The reason that I ask is that every time you start Fusion a new log file is generated. In the ones that you attached, I see that the last you restarted Fusion is on July 27th. I want to confirm if you had a chance to restart Fusion since then?
Also, in your job status (right next to the Help icon on the upper right corner), do you see a "1" indicating that you have pending updates available?
Can you please also try the below:
1. Shut down and close Fusion
2. Go to C:\Users\<username>\Appdata\Local\Autodesk\Web Services
3. Delete file Loginstate.xml
What this will do is the next time you launch Fusion 360, it will force you to login again. See if the save problem is resolved.
I still don't have a root cause yet on why this is happening but from the logs these are the things that stand out immediately.