Victorian lexicographers, as you might expect, present almost gingerbreadlike ornamentations upon shit. An 1857 dictionary features shitesticks and shiterags (both meaning a miser) and the delightful "exclamation of contempt" shittletidee, while a 1875 study notes the institution of Shit-Sack-Day, which seems to involve apples. Shit-Sack-Day, by the way, falls on May 29. I trust you will not confuse it with Shitten Saturday, which is another occasion altogether.
Somehow, among all this shit, I discovered a different previous usage of shit-faced—and for us to connect shit to face, the above history does help. Namely, I refer to the 1825 edition of John Jamieson's Dictionary of the Scottish Language. There are some real gems in there—it is a veritable adder full of precious stones—words like to shog, a verb that means "to shake from corpulence."
But in particular, I draw your attention 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." One might imagine this usage arising late at night, while stepping on children's toys in the dark. But no—this shit, Jamieson writes, is indeed derived from the kittenish chit.
Chit-faced, in fact, already had a long history. Thomas Dekker's 1622 play The Virgin Martyr includes this complaint: "I stole but a durty pudding, last day, out of an alms-basket, to give my dog, when he was hungry, and the peaking chitfaced page hit me int' teeth with it." There's nothing excremental about it—though, admittedly, it's immediately followed by a line about dropping a turd into a bowl of porridge.
All of which gets us no closer to the notion of drunkenness, but it does show that there's more than one way to get shit-faced in Scotland.
But perhaps you already knew that.