It is so old. It is also so '80s-to-present.
There is a usage of so that has become far more popular in the last 40 years, but "I'm so full" would not be an example of it. "So" has applied to adjectives since Old English: so old, so bold, so cold, so beautiful.
The recent usage involves applying so as an intensifier to nouns, other adverbs, verbs, or adjectives that customarily would not take an intensifier. Here's how the Oxford English Dictionary describes it:
a. Modifying a noun, or an adjective or adverb which does not usually admit comparison: extremely, characteristically.
b. Modifying a verb: definitely, decidedly. Frequently in negative constructions.
The Routledge Dictionary of Modern American Slang and Unconventional English, noting the "attitude" and "pronunciation" with which this meaning is delivered, dates this usage to 1979 as used in the film Manhattan:
Oh, please, you know, God, you're so the opposite!
"So" didn't usually modify nouns before this point. Here's an example of so applied to the colloquial phrase "into you":
He's made it clear that he's so not into you that he couldn't even bother to leave you a Post-it.
As for region of origin, the Oxford English Dictionary claims it is "chiefly US," and the first cited instance is from an American film.