Unctuously is the adverb: doing or saying something in that oily, over-friendly, over-praising way that makes the listener uncomfortable. It goes almost entirely with verbs of speaking and facial expression — he smiled unctuously, she agreed unctuously, 'of course, madam,' he said unctuously. Because of that, its natural home is fiction: it is one of those adverbs a novelist hangs on a line of dialogue to tell you, in one word, that the speaker is not to be trusted. Notice that it describes the manner, not the content. The words themselves may be perfectly correct — 'you are absolutely right', 'what a wonderful idea' — and the unctuously is doing all the work, telling you the tone was too warm, too smooth, too eager to please. That gap between polite words and greasy delivery is the whole point of the word. It is formal and literary; in speech people would say 'he was being really smarmy about it'. Underneath is the Latin picture of oil, which is why the word feels slippery even before you know what it means.