Adust comes from Latin adustus, the past participle of "adurere" ("to set fire to"), a verb formed from the Latin prefix ad- and the verb "urere" ("to burn"). It entered the English language in the early 15th century as a medical term related to the four bodily humors - black bile, blood, phlegm, and yellow bile - which were believed at the time to determine a person's health and temperament. "Adust" was used to describe a condition of the humors in which they supposedly became heated or combusted. Adust black bile in particular was believed to be a source of melancholy. The association with melancholy gave rise to an adjectival sense of adust meaning "of a gloomy appearance or disposition," but that sense is now considered archaic.
Middle English "treated with intense heat (of medical ingredients), altered by body heat (of humors)," borrowed from Medieval Latin adustus, going back to Latin, "burnt, scorched," from past participle of adūrere" to scorch, burn up, cauterize," from ad-ad- + ūrere (past participle ustus) "to expose to fire, burn, scorch," going back to Indo-European *h1eu̯s- "burn (transitive), singe," whence Greek heúō, heúein "to scorch" (with presumed metathesis of h), Sanskrit óṣati "(s/he) burns," uṣṭáḥ "burnt up"