I had one of our developers look at this issue with me. We went to the location of the IDLE in the 32-bit Python folder and opened the idle.pyw. When the IDLE window came up, it was still referencing 64-bit Python in the window. We closed that and opened the idle.py file (so that the command window opened with the IDLE) and the version of Python that was running in the command window was 64-bit, so it was calling pythonw.exe in the C:\Python27\ArcGISx6410.1 folder. I'm not sure why the 32-bit IDLE was calling the 64-bit executable or where I could've changed that path, but I just ended up uninstalling Desktop and re-installing it (not ideal, I know), and it seemed to fix it because now the 32-bit IDLE references the 32-bit Python executable.
I also deleted my Python toolbox (because after that it was causing my ArcCatalog to crash before it even opened) and created a fresh one. Luckily, I didn't have much in the toolbox to rewrite and just saved the code in a text file for reference.
Thanks though, Chris, for helping me to dig around a few areas to see what the error could be. Do you know, though, where I could change the path to the Python executable if this happens again so I don't have to uninstall and re-install Desktop? I know that's kind of the last ditch fix to correct references.