Probably too many to mention, but two I can't sing and play along to without crying would be For No One - The Beatles, and '39 - Queen ....heartbreakersFor me, two stand out: "He Stopped Loving Her Today" by George Jones and "Back on the Chain Gang" by The Pretenders.
Which songs "really get to you"?
"It Makes No Difference" - The Band (This version below). When Danko gets to the part "Well, I love you so much. It's all I can do" it's instant goosebumps and misty eyes. Not sure why, his voice and the song just hit me.