The modern sense, with its dominant note of servility, was entering the language in Shakespeare's time. Iago talks of "Many a duteous and knee-crooking knave... doting on his own obsequious bondage" (Othello, I.i.46).
But elsewhere it has only older senses. The oldest, from the 15th century, was "devoted, ready to please", in a positive sense, as when Falstaff compliments Mistress Ford on being "obsequious in your love" (The Merry Wives of Windsor, IV.ii.2) and Angelo talks of the crowds surrounding the king "in obsequious fondness" (Measure for Measure, II.iv.28). But by the end of the 16th century, a sense related to obsequies had emerged, referring to the actions appropriate after a death. Claudius talks of a bereaved son obligated to "do obsequious sorrow" (Hamlet, I.ii.98). Lucius invites Marcus to "shed obsequious tears" over the body of Titus (Titus Andronicus, V.iii.151). The sonneteer also sheds "many a holy and obsequious tear" (Sonnet 31, line 5). Here the word simply means "dutiful".