Does accuracy affect in-hand weapon sway or just how accurate bullets are?

Weapon stability is a hidden stat as it was in Borderlands. Weapon modifiers can improve stability. In fact apparently weapons have "accessories" now (you will see this on the weapon model though you might not recognize it). As with Borderlands, you'll want to take guns out for a spin to see how they handle. I believe can stocks help with stability, so if you see a gun with a stock take it for a whirl. Obviously any scoped weapon will show off stability (or lack thereof), and Sniper Rifles are sort of the kings of this.

Accuracy is a separate stat from stability. I do not know if there is a correlation between the two (and certainly can't speculate on the strength of that correlation).

Some manufacturers were known for making more stable weapons than others though I can't recall off hand who is known for what; keep that in mind. Well, I remember that Tediore makes exploding, teleporting guns.

By the by, this Borderlands 2 Wiki seems the most developed so far. It references stability in several places (including Pixel's link in the comments). It's probably worth pointing out that all of the terms we use may not have the same meanings internally in the engine. Ben, that article about the Accuracy Pool in the comments doesn't seem to contradict what I wrote here, but may well be illustrating how externally we see one model of how things work but it's an abstraction. I don't doubt that stability, recoil, and inherent accuracy interact in strange and mysterious ways in the code!

Accuracy seems tied to sway. Pick up a weapon with a high initial crosshair bloom that increases accuracy as you fire. You'll notice that sway decreases as the crosshairs tighten up.

It looks like sway is a physical representation of crosshair bloom, and weapons with very wide crosshairs will also have high sway, while weapons with very tight crosshairs will have little sway. Test it out.