In Blender 2.90, you can find the "Cycles Hair Shape Settings" in the "Hair Shape Panel" of the "Particle Properties Tab" of the "Properties Editor" ...
... as well as in the "Hair Panel of the "Render Properties Tab" of the "Properties Editor" where you can now choose between "Rounded Ribbons" and "3D Curve":
"Rounded Ribbons" offers you a setting for "Steps" whereas "3D Curves" automatically tries to smoothe it as much as possible.
I haven't tried rendering at a higher resolution yet, but I can bump it up considerably using e-cycles, and see what that does. I'm actually not going to animate it, as it's for a picture book, so higher resolution shouldn't be a problem. Thank you so much, 
spikeyxxx , between you and duerer, it's looking really good! Undoubtedly I'll have more questions before I'm finished, so most likely be hearing from me again.