Basically, I'm getting a bug where the rotations in the animations seem to be skewing the model even though I'm normalizing them. I suspect that the rotations are showing the error, but not causing it, if that makes sense. But that's based on the idea that a normalized quaternion can't skew a model.
Yes. There are two equivalent normalized quaternions for every orientation in 3D space (the unit quaternions form a double cover over the space of orientations), and every unit quaternion can be expressed as an angle-axis rotation.
If you're observing skewing, then it's not from your quaternions themselves, but could be due to non-uniform scales at parent levels of your hierarchy, or a math error somewhere. We'll need to see more of your implementation to advise, though.
Consider asking a question "How do I solve this unwanted skew in my skeletal animation?" instead.