A slight correction to Jim Dixon's transcription of Mordaunt's version:

Mordaunt didn't number the couplets; rather, she just preceded appropriate lines with "Solo:" or "Chorus:" in a running manner.
Each pair of couplets together form one stanza. Between these pairs, the chorus only sings "Royals free--Royals free." It's at the end of each quatrain that the chorus sings the full "Royals free--Royals free, / Studding sails aloft, boys, Royals free." Therefore, the first stanza would be better rendered as:

From Boston Harbour we set sail,
And the wind it was blowing the devil of a gale.
    Royals free--Royals free.
With a Ring-tail set all abaft the mizzen peak,
To see Britannia a-ploughing up the deep.
    Royals free--Royals free,
    Studding sails aloft, boys, Royals free.