"First Time Ever I Saw Your Face". Love at first sight, re-visited in old age—and still just as moving as that first moment, when that other person brought light to "the dark and the endless sky". Up to Eleven in Cash's case as he likely addressed the song to June, recording it not long before her death.
"Love's Been Good To Me". A rover reflecting on his life and despite never belonging anywhere is happy with his life as the love he has experienced was worth it for him.
"Personal Jesus". "Reach out and touch faith", indeed.
"Look At Them Beans". A poor farmer dies. The next year his family is astounded when their land yields a huge crop. Corny, joyous and bittersweet all at the same time (the original version by Joe Tex, "Papa's Dream", is a little heavier on the bittersweet side, while Cash's is more joyous).
"A Boy Named Sue", aside from the comedy, is at its heart a song about a man reconnecting with his father.
"The Baron": An aging pool hustler is challenged to a game by a young man. During the game the old man realizes his challenger is the son he never knew he had. Not only that, but by the end of the game he is reunited with the mother. (It is strongly hinted that the young man orchestrated the entire thing to get his parents back together.) The story behind the song was so strong, it formed the basis of a movie, The Baron and the Kid, starring Cash as the Baron and June Carter Cash as the mother.
The second to last song Johnny Cash recorded (and the last he wrote) before his death two weeks later was "Like the 309". The song by itself isn't heartwarming per se, but it becomes this due to the fact that it is a very funny, grooving song, and shows that, even with his failing health, Cash still had a sense of humor.