Old and ageing -- no, I am not only describing myself here, but I want to look at the history of the words. The noun age came into English from French only after the Norman Conquest; it ousted the Old English word eld or eald. It originally (13th century) referred to a stage of life, and only a hundred years later came to refer to the latter stage of life. The word age has ultimately the same Indo-European base, *ayu-, as the words aeon, ever, eternity, and the Sanskrit Ayurveda. This original Proto-Indo-European (PIE) root form meant life, vital force, or eternity.
Old is a Germanic word. It goes back ultimately to the PIE root *al-, which meant to grow or nourish. Aliment is from the same root, as are alumnus, adult and adolescent.
Adult and adultery, by the way, are not related etymologically. Adultery and adulterate come from a root meaning 'beyond' or 'against'.