Yes, it's in the DT - and with the original words (April, January, June and July), including both verses, too.   The DT doesn't give the original copyright date, though, which was at least as early as 1908.

Here's the blurb I found at YouTube alongside Ruth Etting's recording of Shine On, Harvest Moon."

Shine On, Harvest Moon
(from Ziegfield Follies of 1931
Words and music by Nora Bayes and Jack Norworth
Ruth Etting, vocal. Recorded July 19, 1935, in New York. Originally issued on Columbia 3085D.

This song was first introduced by Nora Bayes and songwriter-husband, Jack Norworth in the Ziegfield Follies of 1908. Ruth Etting's performance of the song in the Ziegfield Follies of 1931 was a tribute to Nora Bayes. The 1931 production of the Follies was the last to be produced under the direction Florenz Ziegfeld, Jr.; he died shortly thereafter in 1932. It's interesting to note that Nora Bayes recorded this song for Victor in 1910 but it was never released.
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The DT does have a MIDI link, but sometimes it's hard to fit MIDI tunes to printed lyrics if you haven't heard the song. YouTube has several postings of singers doing one or both verses of the song, but this one is the one I think is best in terms of both verses being heard loudly and clearly:

Marion Lapham sings "Shine On, Harvest Moon"