Holding down the Option key while you’re dragging will make copies of those files instead of moving them off of iCloud. You can tell that you’re copying files by the green plus sign that’ll appear on your cursor.
To be honest, this may not be a necessary step—Apple claims that files won’t be removed from iCloud during the reset, and they’re being stored locally on your Mac in your ~/Library/Mobile Documents folder anyway—but hey, I’m nothing if not über-paranoid. There should be, like, three umlauts there to indicate how hardcore I am about it.
In any case, after you’re satisfied that your important files are safe, go to iCloud.com and log in. At the upper-right of the screen, you should see your name. Click that and choose “Account Settings.”
On the pop-up that appears, select “Advanced.”
Pick “Reset Documents & Data,” and then you can click the same text on the next screen to start the process. As you can tell from my screenshot below, this reset can only be done once per day.
Apple says that you’ll then need to restart all of your associated iCloud devices, but afterward, you should probably make an edit to one of your iCloud documents and see if the change populates everywhere. If it does, awesome! Go you for fixing your own problem! If it doesn’t, then Apple’s next suggested step is to contact iCloud support. Boo on having to do that, I say. Boo.