The opening line "'Twas a dark and stormy night and the moon was shining bright ...." is found in the Ballad Index. It is listed as Round Roud #13616.

See here: http://www.fresnostate.edu/folklore/ballads/San038.html

Twas in the Month of August In the Middle of July (She Said the Same to Me)
DESCRIPTION: "'Twas in the month of August, or the middle of July, One evening I went walking, a fair maiden I did spy; She was mournin' for her true love, who was in Amerikee, Agh, divil a word I said to her, and she said the same to me!"
AUTHOR: unknown
EARLIEST DATE: 1914 (GreigDuncan8)
KEYWORDS: humorous nonsense paradox separation emigration
FOUND IN: Britain(Scotland(Aber)) US(MW)
REFERENCES (3 citations):
GreigDuncan8 1704, "'Twas in the Month of August" (1 text)
Sandburg, pp. 38-39, "She Said the Same to Me" (1 short text, 1 tune)
DT, SAIDSAME
Roud #13616
NOTES [164 words]: Roud has five references for numbers #13523 (Sandburg) and #13616. Besides Sandburg and GreigDuncan8, he has "It Was on a Month of Sunday" ("It was on the month of Sunday in the city of July"), "On a Cold and Frosty Morning" ("A cold and frosty morning in the middle of July"), and "The 25th of Liverpool" ("On the 24th of Liverpool, in the city of July"), all from recordings. It seems reasonable to me that these are all the same song. Each verse of Sandburg and GreigDuncan8 is nonsense, on the order of "Three Little Girls, A-Skating Went" and "'Twas a dark and stormy night and the moon was shining bright ...." GreigDuncan8 shares Sandburg's first two lines in a first verse "'Twas in the month of August, In the middle of July, The snow was falling thick and fast The weather being dry." The singer hires a tramway car to cross the sea, falls in love with a French girl from a few miles out of Tipperary, and tells about his father "being a dairymaid, Aboard a Sunday boat." - BS
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File: San038