I found a verse in Frank Clune's 1957 "Scandals of Sydney Town",

Goodbye, Sydney Town,
Sydney Town, goodbye!
I am leaving you today
For a country far away;
For today I'm stony broke,
Without a single brown.
If I make a fortune
I'll come back and spend it
In dear old Sydney Town
                   OLD SONG

so I went looking for it, & couldn't find it as it actually 2 songs - one about Sydney, the other about Melbourne which was a bit easier to find (thanks to Rob Willis who has seen both songs over the years)

Fortunately TROVE has the sheetmusic for both songs! Goodbye Sydney Town - words Leonard Nelson, music Fred Hall   Goodbye Melbourne Town - words Leonard Nelson, music Fred Hall

1. GOODBYE SYDNEY TOWN - words Leonard Nelson, music Fred Hall. Allen's popular Sixpenny Songs, no. 78

Lonely and sad stood a brave honest lad
On the deck of a steamer one day,
He was working his passage far over the sea
To England many miles away.
He was leaving Australia the land of his birth,
Where he failed to succeed though he tried,
As the ship moved away he took one longing look
At the city he loved, then he cried.
chorus

Goodbye, Sydney Town, Sydney Town, goodbye;
I am leaving you today for a country far away,
Though today I'm stony broke without a single brown,
If I make a fortune I'll come back and spend it in dear old Melbourne Town

From the ship that was now swiftly sailing away
He watched the crowd saying goodbye;
Some with bright faces were cheering their friends,
Others heaved many a sigh,
Still his thoughts were not with that fast fading crowd,
But the old town he loved, oh, so dear,
As he said with a sigh, I'll come back bye-and-bye,
Once again came the words loud and clear.

chorus

On the pier stood the Dad of that stout-hearted lad,
He'd been saying one last fond good-bye;
As he thought of his boy he could not restrain
The tears that would rise in his eye,
For the ship that was now but a speck on the foam
Held all that made life to him dear,
As he turned from the shore with a heart beating sore,
In fancy these words he could hear :

chorus

~~~~~~~~~~~~~~~~~~~

2. GOODBYE MELBOURNE TOWN - words Leonard Nelson, music Fred Hall. Allen's popular Sixpenny Songs, no. 77

Lonely and sad stood a brave honest lad
On the deck of a steamer one day,
He was working his passage far over the sea
To England many miles away.
He was leaving Australia the land of his birth,
Where he failed to succeed though he tried,
As the ship moved away he took one longing look
At the city he loved, then he cried.
chorus

Goodbye, Melbourne Town, Melbourne Town, goodbye;
I am leaving you today for a country far away,
Though today I'm stony broke without a single brown,
If I make a fortune I'll come back and spend it in dear old Melbourne Town

From the ship that was now swiftly sailing away
He watched the crowd saying goodbye;
Some with bright faces were cheering their friends,
Others heaved many a sigh,
Still his thoughts were not with that fast fading crowd,
But the old town he loved, oh, so dear,
As he said with a sigh, I'll come back bye-and-bye,
Once again came the words loud and clear.

chorus

On the pier stood the Dad of that stout-hearted lad,
He'd been saying one last fond good-bye;
As he thought of his boy he could not restrain
The tears that would rise in his eye,
For the ship that was now but a speck on the foam
Held all that made life to him dear,
As he turned from the shore with a heart beating sore,
In fancy these words he could hear :

chorus

~~~~~~~~~~~~~~~~

I also found this video Goodbye Melbourne Town/Botany Bay/Wild Rover No More (Medley) by Slim Dusty but he only uses the chorus