Lenny Solomon makes Shlock Rock.
Country Yossi does have some touching songs (Deaf Man in the Shteeble, anyone?), but mostly does funny. He also doesn’t write much original music, although he does write his own lyrics.
IMO, Abie Rotenberg is one of the best (if not the best) composer of Jewish music over the last century. Look how many of his songs are known by almost the entire Jewish world! They have withstood the test of time. He also has some funny songs, but I don’t know if he wrote his own lyrics or someone else did it for him. They’re in the Journeys series: Ninth Man on the Team, The Wedding Song, Atheist Convention, etc.
But yes, AR does touching better than most. Some songs were pointed out already. I’d add In a Vinkaleh, Conversation in the Womb, Who Am I?, Memories, and some others.