Frank Sinatra recorded quite a few great songs, but most of them work best as Foxtrots. He had one hit that is clearly a Rumba (Rumba is a "rhythm" dance in American style dancing): "Strangers In The Night."
Another good Rumba is "The Way You Look Tonight" as recorded by Michael Buble. Sinatra has a wonderful version of the song, but it's a Foxtrot. Buble's version is different. (The Buble version is fairly recent, but the song is older. It won the Academy Award for Best Original Song in 1936, as originally sung by Fred Astaire in the movie "Swing Time." Lots of singers have recorded it since then.)
Also, quite a few Elvis hits might work for you. "All Shook Up," "Don't Be Cruel," "Hound Dog," "Proud Mary" all make good Swings. And there are others.