Okay after many hours in 3 sessions of chatting with Adobe support the problem was resolved, and I was able to download InDesign CC. Here's what I had to do:
I had to remove EVERYTHING Adobe from my system. All CS6 apps. All folders anywhere with the word Adobe in the name. Ran the Cleaner Tool. Then launched the Activity Monitor and there were certain processes that I had to turn off. Then, and only then, did I reinstall Creative Cloud and InDesign CC. I haven't tried reinstalling any CS6 apps yet, and I hope that doesn't mess things up.
I recommend having Adobe Tech support walk you through it, because there are so many things that had to be removed from so many places on my mac. It was way more complicated than just uninstalling apps.