late 14c., "condition of being foolish; foolish love, infatuation," literally "the condition of one who dotes," from dote (v.) + -age. Also from late 14c. as "senility; feebleness or imbecility of mind in old age."
c. 1200, doten, "behave irrationally, do foolish things, be or become silly or deranged," also "be feeble-minded from age," probably from an unrecorded Old English word akin to Middle Low German and Middle Dutch doten "be foolish, be out of one's mind," all of which are of unknown origin, or directly from these words.
Century Dictionary and OED compare Dutch dutten "take a nap; mope;" Icelandic dotta "to nod, sleep;" Middle High German totzen "take a nap." Wedgwood writes, "The radical sense seems to be to nod the head, thence to become sleepy, to doze, to become confused in the understanding."
From late 15c. as "be infatuated, bestow excessive love." Also in Middle English "to decay, deteriorate," in reference to rotten timber, etc. (mid-15c.). There was a noun dote "fool, simpleton, senile man" (mid-12c.), but Middle English Compendium considers this to be from the verb. Related: Doted; dotes; doting.
word-forming element in nouns of act, process, function, condition, from Old French and French -age, from Late Latin -aticum "belonging to, related to," originally neuter adjectival suffix, from PIE *-at- (source of Latin -atus, past participle suffix of verbs of the first conjugation) + *-(i)ko-, secondary suffix forming adjectives (see -ic).
Others are reading
Definitions of dotage from WordNet
mental infirmity as a consequence of old age; sometimes shown by foolish infatuations;