Paul Collins traces the roots and cognates of "shit-faced" back to this entry:
SHIT-FACED, adj. Having a very small face, as a child, Clydes[dale].; q. chit-faced?
Instead of, say, a deeply unfortunate drunken pratfall, this shit-faced may come from the old Scottish fondness for referring to children as little shits; Jamieson's 1818 edition notes just such a "contemptuous designation for a child."
The Dish 