The two Carpenter recordings have different tunes, but both put the stress on a-WAY, rather than HEAVE. And interestingly, neither of them have anything like a rolling/ruling king.

Rees Baldwin
Have you seen my Bowery queen
Haul away, heave away
Have you seen my Bowery queen
For we are bound for South Australia

Heave away, haul away
Heave away, haul away
For we are bound for South Australia

Way down south, where I was born
Heave away, haul away
Among the fields of yellow corn
For we are bound for South Australia

note that in verse 1 it is "Haul away, heave away", while in Verse 2 the order is reversed.
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J S Scott

Heave away, my lowlands below
Heave away, heave away
Heave away, my [dirty loving Flo?]
Cos we're bound for South Australia

Oh South Australia is the place for me
Heave away, heave away
We [work?] all the money and we goes on the spree
And we're bound for South Australia

Heave away, my lowlands below
Heave away, heave away
Heave away, my [dirty loving Flo?]
Cos we're bound for South Australia

There's a nice [get in] on Australia's shore
Heave away, heave away
I'm not going to sea no more
And we're bound for South Australia

Heave away, my lowlands below
Heave away, heave away
Heave away, my [dirty loving Flo?]
Cos we're bound for South Australia

much of Scott's singing is very unclear, but it is definitely all heaving, and no hauling