Elise,
There were at least two newspapers published in the mid/late 19th century. One was the COMET (democrat) and the other was the STAR (Republican). Also, news of the Lafourche parish area could have appeared in any of the newspapers of the surrounding areas - Assumption, Ascension, St. James, St. John, St. Charles and, of course, Terrebonne parish.

The real questions: Are there any existing copies of the earlier papers? Where are they?

LSU Special Collections is the repository for many older newspapers under the "Louisiana NP", sub-project of the "U.S. Newspaper Program", an NEH funded program.

You can also contact the Terrebonne Genealogical Society (TGS) for information on the local newspapers. They have extracted and published information from early newspapers in their quarterly.

Oh yes, news of the rural parishes could also have appeared in early New Orleans papers, but I don't know if that would have been a regular section. Everyday happenings in Lafourche parish may not have been particularly newsworthy.