The earliest mention I find to "Rio Grande" in my notes comes from,

1868 Dallas, E. S., ed. "On Shanties." _Once a Week_ 31 (1 Aug. 1868).

(The author of this article is unknown.)

Part of the chorus is given, in the following passage:

There is an air of romance about California, the Brazils, and Mexico, that has a peculiar charm for Jack, and has made them the subject of many a favourite shanty, as Rio Grande, Valparaiso, Round the Horn, and Santa Anna...

Rio Grande is perhaps the greatest favourite of this description of songs, but all the beauty lies in the mournful air :—

To Rio Grande we're bound away, away to Rio ; 

Then fare you well, my pretty young girls, 
   
We're bound for the Rio Grande.

The song next turns up in,

1869        Payn, James, ed. "Sailors' Shanties and Sea Songs." _Chambers's Journal_ 4(311) (11 December 1869): 794-6.

The author of this is also unknown. However, either it is the same author re-using his/her material, or it is directly plagiarized from the previous. The description of "Rio Grande" is practically verbatim.

There is an air of romance about California, the Brazils, and Mexico, that has a peculiar charm for Jack; and he has made them the subject of many a favourite shanty, as Rio Grande, Valparaiso, Round the Horn, and Santa Anna. Rio Grande is perhaps the greatest favourite of this description of songs, but all the beauty lies in the mournful air:

To Rio Grande we 're bound away, away to Rio; 

Then fare you well, my pretty young girls; 
   
We're bound to the Rio Grande.


Without going too crazy here in the analysis...one can see that although the second passage was copied from the first, it changed the chorus from "bound FOR" to "bound TO." This helps in identifying later writers who used one or other article as a source. Basically it was the second article -- which is the source of OED's early mention of "shanty" -- that was influential. Notably, the writer of the second article may not have known anything of substance about chanties, seeing as how he was only culling earlier articles and, for instance, did not correct the (presumed) mistake of listing "Valparaiso" and "Round the Horn" as 2 diff. items.