IMHO , there aren't standout onsen in Fukuoka City, and if I am being honest, in Fukuoka Ken. You really need to go to Saga, Oita , Kumamoto , Nagasaki-Ken. The onsen in Dazaifu , Funagoya , Hiko-san areas (and there are others) are not that great and are more super-sento style. As always YMMV.