Teenage girls on buses in east central Scotland are often to be heard uttering “Shitey buggery fuck!”
(Put that in your Brigadoon stereotypes and smoke it!)
I should think that the correct adverb for the US as well as Britain would be “shittily.” So, “They treated me really shittily.” But: “That’s okay, because their service was shitty.”
I’ve always seen it as a milder form of fuck.
Shite for me (and others I’ve heard use it) works best if you say it by itself, shit works better after an “ack” or “ahhhh”.
My mates and I thought it a nice touch to have Richard Gere’s character use the word in “The Jackal.” Made him much more Irish than any accent.