I find Lee Hyori’s Don’t Cry, ICON’s Hello, Melody Day’s Another Parting and Akdong Musician’s Melted some of the best “crying my eyes out”-songs right now.
Also try Shinhwa’s Destiny of Love and Hurts. Both (but especially Destiny of Love) are super touching and sung with such emotion that you will almost definitely cry (and\or have to repeat it over and over again).
Smoking (hot) Junjin♡
That look in the 3rd gif is killing me!