South Australian history (non-native) begins with the first shipload of settlers in 1836. It did not become important until later; the first mention of shipping from there is from c. 1850, when timber was sent out for the gold fields in New South Wales.
(Wiki and other online references).

Nine ships were involved in moving the first c. 650 settlers to South Australia (no convicts), I haven't found any songs about them.

Huntington, "Songs the Whalemen Sang" found many songs in ships logs and diaries. Some are sea songs from the the 1850s-1860s or earlier, but many were songs popular on land.

"Rolling Down to Old Mohee," Atkins Adams, out of New Bedford, 1856, is a great song and a great find by Huntington.