"God Only Knows", written by Brian Wilson (music) and Tony Asher (lyrics), performed by The Beach Boys, vocals by Carl Wilson.
"What Becomes Of The Broken Hearted", written by William Weatherspoon, Paul Riser, and James Dean, performed by Jimmy Ruffin. Joan Osborne has also recorded a great version.
"The Weight", written by J.R. Robertson, performed by, of course, The Band.
"No Time To Cry", written and performed by Iris Dement. An incredible song also recorded by Merle Haggard, Kasey Chambers, and Joe Nichols. Iris’ version remains the best.
No Lennon & McCartney songs? They imo were a good writing team (not so good without each other), not a great one. Some disagree with that assessment ;-) .