The badge is misleading. While it says you must complete the game, you must actually complete ALL LEVELS. You cannot use the 2 skips the game provides you with.
exequy, equinox, quixote, exequial, exequies, quadplex, quincunx, quixotes, quixotic, quicotry, equinoxes, exchequer, exquisite, quixotism, exchequers, exquisites, quadplexes, quincunxes, quicotical, quicotisms, quixotries, exquisitely, quincunxial, quixotically, exquisiteness, exquisitenesses
These are the Q+X words that are known to exist.
It be awesome if there was an option that prevented weapons from switching automatically. I am using the frying pan because I prefer it, and it should force me to scramble for the 1 key as soon as I run over a sun flower.