The movie you describe is "Hollywood Hotel", released in 1937/38.
In this movie a saxophonist wins a talent contest and travels to LA to
take up a contract with a Hollywood film studio. He gets fired, and
finds work as singing waiter in a drive in, where he is spotted by a
director of the same studio that fired him.
A more detailed plot summary is available on the Internet Movie
Here are further details of the movie, from the same site:
By the way, Ronald Reagan had a bit part in this film as a radio
The poster is offered by AllPosters:
The soundtrack is offered on CD by Amazon:
The movie is available for hire on 16mm film from swank.com:
The movie is offered on video by TVideo:
Ted Healy's site has some MP3 soundclips from the film:
I started with some Google searches, which did not yield the answer:
"behind the 8 OR eight ball" movie
"drive-in" "fast food" movie +la OR "los angeles" singing
Next, I searched the plot summaries on the Internet Movie Database
with numerous search strings, including:
arrives in LA
arrives in Los Angeles
The phrase "singing waiter" led to "Hollywood Hotel".
Researching this question has piqued my interest, and I'll be looking
out for the next TV screening of this movie!