As a native English speaker, I can tell you that craziness is definitely a more common word. You're right that craze can refer to popularity, whereas craziness refers to insanity.
Craziness is defined as
- The state of being broken in mind; imbecility or weakness of intellect; derangement;2 insanity 1.
- The state of being broken down or weakened; as, the craziness of a ship, or of the limbs.2
- The result or product of being crazy. 2
So craziness refers to someone's insanity. It could also refer to a ridiculous idea or plan. For example,
Person 1: Shall we just head straight for the finish line?
Person 2: That's craziness! They'll see us coming a mile away.
Craze is defined as
- Craziness; insanity;3 an insane condition4
A popular or widespread fad, fashion, etc.; mania:
The newest dance craze4
So craze can refer either to insanity or to a new trend or popular fashion.
I think craze can be used to mean "craziness" or "insanity" if you explicitly mention that the said 'craze' is in and of a particular person. You could say that someone is filled with craze or afire with craze to emphasise their craziness. I would argue that either term is valid for literature and it really depends on what kind of ideas you want to create around the character.
Perhaps worth noting is the adjective form of craze: crazed, which just means 'insane'. If somebody is insane, you could say that they are crazed; in a state of craze. An example that springs to mind is a Hearthstone card called crazed alchemist.