The Oxford English Dictionary has a cite for “US” from 1834. It also notes the “USA” is a variant on “US”, but it doesn’t record cites for “USA” independently. The implication is that the 1834 cite suggests that the “US” abbreviation appeared first, with “USA” coming along some time later, but I don’t know how much later.
The United States Army wasn’t established under that name until 1784. Prior to that you had the “Continental Army”. So if this button is from the Revolutionary Wars, USA is unlikely to stand for “United States Army”.
“USA” might stand for United States Army, and I don’t know for sure if that button I linked to is authentic, but if it’s not, it’s a reproduction of an authentic button. They were called “Intertwined USA buttons”, and they appeared on Continental Army officer uniforms from 1777-83. Here’s a picture of another one from the website of the Colonial Williamsburg Foundation:
From William Emerson’s "Encyclopedia of United States Army Insignia and Uniforms:
It is not my intention here to debate the origins of the abbreviations “US” or “USA”, but a short history of the army’s use of the letters “US” is instructive. During the American Revolution the Continental Army impressed the letters “US” on gun locks, cannon tubes, files and other pieces of ordinance and and equipment to identify them as property of the United States. On the only “insignia” of the period, soldiers’ solid pewter buttons, there were three raised and intertwined letters “USA”. Some officers wore thin metal button shells, crimped over bone or wood disks, bearing raised and intertwined letters “USA” with the S considerably larger.
Well, according to the book that I linked to, and which Captain Amazing also quotes, the button design switched from a “USA” mongram to a “US” monogram in 1808. So most likely the button concerned is either a pre-1808 button or a later reproduction of a pre-1808 button.
In any event, regardless of the provenance of that particular button, the book seems to be authority for the fact that the abbreviation “USA” was in use from at least 1776. And while some have suggested that it stood for “US Army”, the facts are that the institution concerned was call the Continental Army at the time, and that the official name of the Union was “United States of America” from 1776 onwards. So the mongram is, on the whole, more likely to have stood for “United States of America”.
Well, this is hardly very surprising. Until you conceive of the new political entity, you don’t need a name for it. Until the colonies decided that they weren’t going to be colonies any more, they were just “the colonies” (or “the American colonies”, if disambiguation was needed).
Why would we expect the term “United States of America” to be in circulation before the project to establish the United States of America was actively under way? What could such the term have referred to, much before 1776?