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So, I just downloaded 0.14. I started with, but haven't played since 0.11.

 and the save files aren't showing up in the 0.14. For every other game I've played from itch.io I could just download the latest update, delete the old one, and pick up where I left off (with the one other exception being the last one I updated). 

Did the way the save files work change? Do I need to manually move the save files over or something?

There should be no change to the save files. All of my saves still show up, even from v0.1. You should be able to do like you said with deleting the old one. If you're on windows, your saves should be in %appdata%\RenPy. GnB's saves should be in the folder GlitchesAndBitches-1641348167. If you don't see that folder, your saves might have been deleted.