Is it possible? I'm afraid to try since there's no apparent save slots and only the option to Continue. Will it save over my first game if I start and save a new one?

I'm going to say yes, I've seen a couple screens that had the option to choose your save file (like the co-op screen).

But maybe if you make one more thread, the answer will pop into your head magically.

When you make a new character it will open a new save slot.
Next time you click "continue" it will show your different heroes on fate cards.
You can have up to 6 characters at once.