I know you probably don't remember these, they're all from the 50's or 60's. I don't know about any tears, but they all deal with death.
I remember 2 that were actually more sad, There was "Honey" by Bobby Goldsboro, about a man losing his wife; "Don't Cry Daddy" by I think Elvis, but I'm not sure, about a little boy comforting his Dad when his Mom died, they are really sad.
These are not so sad, but all deal with death. There are the car crashes, etc. "Teen Angel", "Leader of the Pack", "Last Kiss", "Tell Laura I Love Her" , "Dead Man's Curve" (Which was ironic because the singer, I think named Dean Torrance-Jan and Dean anyway-, had an eerily similar accident in the same place as the song and survived, barely)"Green, Green Grass of Home" by Tom Jones was a convicted man who was going to be executed. "Tom Dooley" was also about a condemned man who had killed somebody.
There were others-Patches (by Dickie Lee, a later unrelated "Patches" was by somebody else in the 70's) which was suicide. Another that dealt with suicide was "Moody River" by Pat Boone. In "Running Bear Loves Little White Dove" two Native American lovers from warring tribes jump into a river and try and swim to each other and both drown. There were a lot of songs like that in the early '60's. I no doubt left out a few, there were so many.