"Sailors Frolic, or: Life in the East," is the title of a broadside in the Bodleian Library which has the lyrics found in the "Sea Chest."
This broadside includes one more verse before the final 'So all you jolly sailors.'

At the famed Old Barley Mow,
hailed a frigate tight;
Steer'd away, without delay,
and boarded her that night,
She took my watch and money too
and clothes without delay.
Two bullies stout they turned me out,
in Ratcliff Highway.

"Sailors Frolic" is a different song from "Ratcliffe Highway (Rolling Down Wapping)." A different title is needed to avoid confusion. "Sailors Frolic (Life in the East)" is the broadside title, but a better one may be found. "Sailors Frolic" is not in the Traditional Ballad Index. I don't have Roud.

The English Folk Dance and Song Society issued a cd "Rap-a-Tap-Tap" by Frank Purslow in 1960 with "Sailors Frolic;" I don't have it so don't know what tune he used.

One minor correction to the text from "The Sea Chest;" in the second verse last line, figure heads should be plural.

Bodleian Library, Firth 13 (213), no date; Harding B15 (270b); both from the same press. No date or printer given, but the ca. 1840 date in the "Sea Chest" seems about right.