Okay, so it's not specifically a full moon song, but Sweetheart's Holiday does make mention of the man in the moon and, being the Valentine's month, I figured it'd make a good piece of music for the occasion.
The lyricist Irving Kahal worked in vaudeville until his death in 1942 and his song You Brought a New Kind of Love to Me would eventually be featured by Frank Sinatra. On looking up Kahal's information I found he'd written two old songs that have become favorites of mine: Wedding Bells are Breaking Up That Old Gang of Mine and When I Take My Sugar to Tea.
J. Russel Robinson has a closer connection to me, though, being born here in Indianapolis, Indiana where he started composing ragtime with his brother. Robinson was known for his blues influences and in 1918 he joined W. C. Handy's publishing company (Pace and Handy) where he provided new arrangements of popular tunes and even played piano accompaniment for recording sessions. Among Robinson's innumerable compositions are a few real gems including Cab Calloway's 1932 Reefer Man, Meet me in No Special Place (and I'll be There at No Particular Time) which was recorded by Nat King Cole, St. Louis Gal recorded by Bessie Smith, Robinson would eventually die in 1963 of cancer in Palmdale, California.
The following is a fairly common version of Sweetheart's Holiday recorded by Lee Morse. Hopefully it puts you in the mood for love in the light of the full moon!