The "plushiness" comes from Henry's Attic Pigtail cotton. I've attached a picture of it from here. They sell it, but it costs $13.60 for a 1/2 lb.
I've never seen another cotton yarn like this before -- but, then again, I haven't really looked, either!
The Alpine Cotton that is used as the "base" yarn would be fairly easy to find a substitute for, but I don't know where to find something like the Pigtail.