First, some housekeeping: there's a few Tennessee Waltz pages. All of them are rather old, and some of them entirely unproductive. But below is the three of them. I'll leave it to some mudelf to decide how to handle them.

Tennessee Waltz / Rose of Tralee-Are they related?
Tennessee Waltz/Tennasee Waltz (nothing productive came of this thread)
Origins: Tennessee Waltz

And now, some research. From the Sapulda (Okla) Herald, April 29th 1979, p. 3D (link) comes the following interview with King about the song. The quotes are a bit sloppy on the original (not to mention the incredibly choppy nature of the line breaks in the columnized print), but I've left them as-is:

Few people know the history of "Tennessee Waltz," and it required a bit of a hard-sell to convince Pee Wee (he permits no quotes around his nickname) that it should be aired at this time.

"Remember years ago a new theme song became our theme was 'When It's Nighttime In Nevada I'm Dreamin.' (audio) It was an ASCAP tune, and we couldn't play it on the air anymore.

"I had to find something that sounded like it so I wrote this melody.

"It didn't have a name and it didn't have any words. I recalled it "The No Name Waltz."

If you listen to 'Nighttime' and the 'Tennessee Waltz' there is a similarity."

King says "Tennessee Waltz" was born one dark night when his luggage truck and the staton [sic] wagon carrying his band sped toward Nashville from a personal appearance in Texas.

"Redd Stewart (vocalist) and I were in the luggage truck. We were on the Grand Ole Opry at the time, and while traveling were listening to the radio They had just played Bill Monroe's record of 'Kentucky Waltz.'

"Redd said, 'That's terrible. We make our living in Nashville, Tenn., and we live in Nashville, Tenn., and nobody has written the 'Tennessee Waltz.' "

Stewart had an unlighted cigar in his mouth, and reached into the glove compartment for a pack of book matches. When he lighted the cigar, he looked casually at the cover of the pack of matches.

It showed a man and woman, in silhouette, waltzing.

"We'll write the words and call it the 'Tennessee Waltz,' " Stewart said.

"Notes were made on the match cover," Pee Wee recalls. "I only wish we had kept that match box cover.

"Redd had the lyrics from that night, and I had the melody. We put it together and only changed one line.

"The middle part of it. 'I remember the night and the Tennessee Waltz,' it was originally written: 'Oh, the Tennessee Waltz, the Tennessee Waltz, now I know how much I have lost.' "

King gives full credit to a girl from Oklahoma[,] Patti Page, for reviving "Tennessee Waltz" and sending it on a record popularity spree that continues today. But Pee Wee will never be able to play it again on his world-famous accordion.

The 74-year-old member of the Country Music Hall of Fame says those days have ended as the result of a recent stroke.

I looked up Pee Wee King's track credits on the most reliable country discography site I know of (Praguefrank) and found no evidence a recording was made, radio or otherwise, of his performance of No Name Waltz. However, I did find the following information from a 1970s interview.