Songs Where Artists Sings Their Own Name in Song?

Bo Diddley https://www.youtube.com/watch?v=dLcYuuljrD4

In David Lee Roth’s version of “Just a Gigolo/I Ain’t Got Nobody” at one point he modifies “people know the part I’m playing” to “people know the part Dave’s playing”.

I had to do a little digging to make sure that the "Shirley" is her, and indeed it is:
"'Shirley'' Music written by Alvin Leroy Homes, and words by Al Stillman. Released 1966 on the album I’ve Got A Song For You and on a single...This song was specially written for Shirley Bassey, and a firm favourite in her live stage performances over many years."

https://www.youtube.com/watch?v=lUtnDGwbv6M

Jerry Lee Lewis would do that quite often.

"They tell me I look like Merle Haggard, and sound a lot like David Allan Coe."

David Allan Coe - Longhaired Redneck

"Well, I've heard my name a few times in your phone book. And I've seen it on signs where I've played. But the only time I know, I'll hear 'David Allan Coe,' is when Jesus has his final judgment day."

David Allan Coe - You Never Even Called Me By My Name

Does the title count?

There's the Ballad of John and Yoko, but I don't think John Lennon says his own name in the lyrics, though it's written in the first person.

In ‘Crazy Little Thing Called Love’ by Queen, they all say ‘Ready Freddie’ about 3/4s of the way through.

In 1988, Chubby Checker teamed with The Fat Boys and rerecorded The Twist.
They sing their names too.
https://youtu.be/bqsQvaBCARE?si=NC9WIlNrb5hYNKKD&t=93