I just got off the phone with Apple Customer Support, who got iWorks working with iCloud again in less than 15 minutes. To troubleshoot, they first had me create a new user account on my iMac Computer. Once logged in to that new computer user account, I set it up to access my iCloud docs. iWorks accesssed iCloud on this new user profile with no problem. We logged back into my user account, where it (maybe not surprisingly) still didn't work.
Next, to try and fix the problem in my normal user account, we opened the Finder, held the option key down until the "Library" option appeared, clicked on Library, opened the "Application Support" Folder and looked for a folder named "Ubiquity". We deleted the file and restarted the computer; iWorks now accesses the iCloud without a problem. 🙂 🙂
The support technician said that this kind of problem can happen when files aren't completely uploaded on one device, subsequently opened on another and edited, creating a file conflict. The same thing has happened to me with Dropbox a few times, especially since mutliple users can access some of my shared files. Unlike Dropbox, Macs store the iCloud sync conflict in a folder as sort of an "unfinished conversation", and the program can't sync as long as that unfinished conversation is hanging around. He said any kind of syncing can cause this type of issue, but iCloud syncs should get better over time.
In the meantime, I'm going to try and be much more diligent about making sure files are completely synced before closing apps on both my Mac and iOS devices.