Stevie does something similar in “As” (at about 3:45).
Ray Davies switches vocal styles in at least a few Kinks songs, like “The First Time We Fall In Love” and “She’s Bought a Hat Like Princess Marina” (though that one may be more an example of the song itself switching gears a couple of times).
David Bowie seems to be voicing several characters (members of the Spiders from Mars) on “Ziggy Stardust”, especially on the first line of the second verse (“Ziggy really sang…”).
OK then, check out Hello Tosh by the aforementioned Pato Banton, where he’s himself, his mother, and a doctor, or Don’t Sniff Coke where he’s himself, his mother, Roadie, who works at P.A. Hire, and even changes his own voice when talking about Sensi, the healin of the nation.
Would Frankie Valli going into falsetto count? Based on your criteria I wouldn’t think so.
Speaking of Bowie, he kinda tries to do this in “Space Oddity”, alternating between Major Tom and Ground Control…not a whole lot of difference to my ears.
At least the countdown “voice” sounds different.
Roger Daltrey sings different voices (and characters) in Tommy. “Go to the Mirror” (the Doctor and Tommy) and “We’re Not Gonna Take It” (Tommy and his disciples) are clear choices.
Daltrey also sings two character voices in “A Quick One While He’s Away” (not counting harmonized sections).
If that were so, anyone who yodels.
Would Paul McCartney’s “Uncle Albert Admiral Halsey” be an example of what you’re looking for?
My problem is, I can’t be sure if it’s Daltrey doing two different voices, or Daltrey and Townshend.
Robert Plant (in his Led Zep days at least) would be an example of a singer who switches between vocal styles within songs on a regular basis.
Oh yeah, and Jim Morrison.
Stewart sings the intro. He also wrote the song.
Alice Cooper - Years Ago
I’m a little boy
No, I’m a great big man
No, let’s be a little boy
For a little while longer
Maybe an hour?
I don’t know why, but the first song that came to mind was Troglodyte by the Jimmy Castor Bunch.
“I’ll sock it to ya, Daddy!”
Yeah, Ray Davies changes inflections often in “Dedicated Follower of Fashion.” Like in the following lyrics:
He flits from shop to shop just like a butterfly.
In matters of the cloth he is as fickle as can be,
Cause he’s a dedicated follower of fashion.
It’s a great song and he’s most definitely adopting a gay stereotype accent in places. Not that there’s anything wrong with that.
I think this counts: You Know My Name, Look Up the Number–The Beatles
I’ve always wondered what exactly Mick Jagger was on (besides Helium) when he decided to sing Emotional Rescue that way! 
Trying to cash in on the disco era. Everyone knows “Some Girls” is the last Stones album with any relevance. Talk about crash and burn.
Jagger goes from falsetto during the chorus to typical swagger rapping in “Fool To Cry”. But unless you mean someone who has altered their voice for the duration of the song, “If I Was Your Girlfriend” by Prince would be my first thought. He modulated his voice to a higher pitch for “The Black Album” and “Sign O The Times”.
Jarboe on “Hypogirl” - ordinarily her voice is somewhat more mellow, but this is just terrifying.
Not entirely sure it counts, but Dylan’s voice on the Nashville Skyline album (example) sounds far different from the way he usually sings.
At the risk of being obvious, I will nominate, “The Bohemian Rhapsody” and “Stairway to Heaven.”
I assume the OP isn’t really looking for something like Jagger on “Dear Doctor” where at one point the song’s protaganist is quoting a (Dear John) note from his (now ex-) fiancee, and Jagger does an almost-spoken female-type voice (with an attempt at a hick U.S. accent).