Thanks for the idea - unfortunately it didn't work, and the plot thickens. So I tried all of that, deleting my saves, starting a new game, saving and quitting that, *deleting that new save I just made*, then copying all my old saves back in. When I started the game again, it still saw just that lone new save I had just made even though I'd deleted it!
On top of this, the game still shows all my Stats and Medals from the previous 10 hours.
I have all my saves locally (allegedly), but there are none in the Steam Cloud, and I can't figure out how to force them up there. Everything I've tried hasn't worked.