If it's a correct, non-idiomatic usage, is "possessed" an adjective, or...?
What is "of" under that circumstance?
Possessed of as a unit is classified as an adjective meaning possessing, which Dictionary.com calls an idiom. Of is a preposition added in order to make the idiom. Compare it with I am possessed by impeccable grammar: this makes use of the normal meaning of possessed. Possessed is usually the past participle of possess (so it usually means owned). However, when you add the preposition of, possessed of no longer means owned, it means owning.
So I am possessed of impeccable grammar is interpreted I am possessing impeccable grammar (idiomatic interpretation), rather than I am owned by impeccable grammar (normal interpretation, overridden by the idiom).
"Possessed of" means "own" or "have". So "I am possessed of impeccable grammar" means "I have impeccable grammar".