The term 'annoyingly insensitive' can be described by the crossword solutions of OBTUSE, PRUDE, WHINES, POKY, PULED, TEASE, and PESKY. OBTUSE means lacking sharpness or quickness of intellect, while PRUDE means excessively modest or proper. WHINES refers to a high-pitched complaint or cry, and POKY means slow-moving or sluggish. PULED means to whine or whimper, and TEASE means to provoke or annoy. Lastly, PESKY means troublesome or annoying. Together, these words paint a picture of someone who is not only slow-witted and overly proper, but also prone to complaining and provoking others in an irritating way.