I've just been checking through my archive of old 78 recordings of sea music. I haven't turned up much and none of it rivals what Desert Dancer and Q have dug up:

Richard Dyer-Bennet recorded on Richard Dyer-Bennet - Lute Singer - Ballads and Folk Songs, Keynote Recordings, Album No. 108, © 1941:

Hullabaloo Belay
What Shall We Do with a Drunken Sailor

Leonard Warren (with Orchestra and Chorus!) recorded on Sea Shanties, RCA Victor Red Seal Records, circa 1940, collected and arranged by Tom Scott:

Blow the Man Down
Rio Grande
The Drummer & the Cook (identified as a foc'sle song rather than a shanty)
Shenandoah
Haul-A-Way Joe
Low Lands
The Drunken Sailor
A-Rovin'

Cheerily,
Charley Noble