I apologise. I should have checked my broadside indexes straight away.
'Cheer up, Sam' was printed on broadsides by Williamson of Newcastle 1856-67, Pearson of Manchester, Sanderson of Edinburgh and Hodges of London and there is a broadside at the Bodleian Broadside website Firth b27 (179) all about the same period, certainly none predating 1850 so it looks like the American Memory sheet music is probably the original.
In my shanty index no texts turn up under that title.

Re 'Heave to the girls'. I also don't think anyone would quote that part of the 'Rio' versions. In effect its meaning crosses two lines,
We'll sing as we heave,
To the maidens we leave.
It distorts the meaning if you take out the said phrase.

A long shot....a distortion/oral version of 'Here's to the maiden of bashful fifteen' taken to sea? 'here's to the girls'.

Another suggestion...toasts of that nature were very common in the early 19thc and some of them were in the form of a sung verse. Could 'here's to the girls' have been misheard as 'Heave to the girls'??