I'm surprised that nobody (including me) has mentioned the singer who has a whole litany of songs with both singing and talking.  Here are just some of the hits by Bobby Bare that demonstrate his narration skills...
"Shame On Me"
"Detroit City"
"500 Miles Away From Home"
"Miller's Cave"
Of course, he got an early start with his hit "All American Boy" back in 1959.
 

The first one I thought of, even before I saw Leader Of The Pack on the list
I Can Never Go Home Anymore - The Shangri-Las
 
You Never Even Called Me By My Name - David Allan Coe
Supercalifragilisticexpialidocious - Julie Andrews & Dick Van Dyke
Along Came Jones - Ray Stevens
Que Sera Sera -Doris Day
 
It qualifies but it’s not really a “song”
Topsy Part 1 & 2 - Cozy Cole
 
Maybe I shouldn’t bring these up
Don't Go Near the Indians - Rex Allen
Don't Go Near The Eskimos - Ben Colder
 
 
While trying to think of these, I kept coming up with songs I couldn’t decide if were sung, spoken or a combination. Maybe there should be a new thread for these
Skip A Rope - Henson Cargill
Little Boxes - Pete Seeger
Jimmy Dean - Big John
The Devil Went Down to Georgia - Charlie Daniels
Slow Poke - Pee Wee King
Tennessee Waltz - Patti Page
Where Will The Dimple Be? - Rosemary Clooney
Something Stupid - Frank & Nancy Sinatra
16 Tons - Tennessee Ernie Ford
Wildwood Weed • Jim Stafford
Five Feet High and Rising · Johnny Cash
When You're Hot, You're Hot · Jerry Reed
Hey, Good Lookin' · Hank Williams
Heartaches By the Number · Ray Price
Daddy Sang Bass · Johnny Cash
Sink The Bismarck · Johnny Horton
Amos Moses · Jerry Reed
Me And Bobby McGee - Janis Joplin
Look What They've Done To My Song, Ma – Melanie
The Night They Drove Old Dixie Down - Joan Baez
Before The Next Teardrop Falls - Freddy Fender
The Rose · Bette Midler
Brand New Key · Melanie
The Night the Lights Went out in Georgia - Vicki Lawrence
My Ding A Ling - Chuck Berry
House Of The Rising Sun - The Animals
Monday, Monday - The Mamas & The Papas
Different Drum - Linda Ronstadt
Tennessee Bird Walk  - Jack Blanchard and Misty Morgan
May The Bird Of Paradise Fly Up Your Nose - Little Jimmy Dickens
Hello Walls - Faron Young
Amarillo By Morning - George Strait
Stand By Your Man - Tammy Wynette
He Stopped Loving Her Today - George Jones
Louisiana Woman, Mississippi Man - Conway Twitty & Loretta Lynn
Hot Rod Race - Jimmie Dolan
Hot Rod Lincoln - Charley Ryan
Little Green Apples - O.C. Smith
  
And the king of these (and of the road) - Roger Miller
You Can't Roller Skate In A Buffalo Herd
Do-Wacka-Do
Chug-A-Lug
Dang Me
Engine Engine #9
Kansas City Star

How about the opposite? Spoken word with song passages
The Flying Saucer Part 1 & 2 Buchanan & Goodman

and another
The Blob - The Five Blobs
 
And now I officially hate Roman, I can’t stop thinking about these songs.  At this point, I’d love a good earworm.

Every time I walk away from the computer, I think of another
 
99 Luftballons - NENA

Lorne wrote:

Let me see if I can help out here as well ... I think that this is the perfect song for your situation. 


 

too late

Drinking Wine Spo-Dee-O-Dee - Jerry Lee Lewis
The Streak - Ray Stevens
The Ballad Of Irving - Frank Gallop
Hello Muddah, Hello Fadduh - Allan Sherman
The Ballad Of Jed Clampett - Lester Flatt ,Earl Scruggs
Little Egypt - The Coasters
Charlie Brown · The Coasters
Stager Lee - Lloyd Price
Alley Oop - Dante & the Evergreens
Abba Dabba Honeymoon - Debbie Reynolds & Carleton Carpenter
Mr. Custer - Larry Verne
 

A few more
 
Don’t know why I didn’t think of the first one on the list sooner
 
Love is Strange- Mickey & Sylvia
Duke of Earl - Gene Chandler
Mr Blue - The Fleetwoods
Speedy Gonzalez - Pat Boone
Let’s Dance - Chris Montez
Fire - The Crazy World Of Arthur Brown
 
One word, no song
Tequila · The Champs

Well Taz, I must say, I think you had a great time with this thread!   I can't believe all the songs that apply.