Integrity is a big word. There's artistic integrity that has to do with the way a song is concocted. Integration into a musical show is one way. Songs of humanitarian issues, another. Jazz integrity is another. A teaching moment in a song can sometimes become didactic or phony. Preachy songs should be avoided because young people can spot them a mile away.   Put songs into continuity with historical events that are of a teaching moment nature.

I think as a songwriter, Woody Guthrie had integrity. I sense a kind of integrity with Steve Earle when he doesn't get off onto this Townes Van Zandt thing.

For me, I would avoid songs that are religious, bordering on preachy, narrow in their view and application, but that's me.

Here's some:

"Black Water" by Jean Ritchie about strip mining.
"Ludlow Massacre" about the tragedy of the history of unions.
"Last Night I Had the Strangest Dream"..Ed McCurdy's ironic comment on the futility of war
"Talking Atomic Blues" by Verne Partlow, a satirical look at the atomic and now nuclear industry
Most all of the songs by Tom Lehrer and Tom Paxton...highly integrative people with an important teaching device....humor.
"Strange Fruit" about the ugliness of race prejudice
"We Shall Overcome" because of the history of the Civil Rights movement.

Any songs that depict social history and comment on it.

"Inch By Inch" ....David Mallet's "hit" about the environment.
Woody Guthries Dustbowl ballads depicting the plight of the Dust Bowl Refugees

Songs in different languages that have a social message.

"Follow the Drinking Gourd" about slavery in the U.S.

Oddly enough, 5 to 11 year olds get these songs. I know because i have sung them for these ages and they are responsive if you explain them right.

Too many people write off the intelligence of younger children in their ability to grasp complicated subjects.

Oh yes, "It Could Be A Wonderful World"...Hy Zaret and ? "Little songs on Big Subjects" is the name of the compilation, I think.

"Two Brothers" and "The Blue and the Gray" to explain the tragedy of the American Civil War.