Before any other answer is given, it must be noted that there is a severe problem with the premises, i.e., that the US changed. In reality, the US stayed the same while the rest of the English speaking world changed. We can see this both in real-life letters and in the fictional letters of epistolary novels.
For example, in the first North American novel The History of Emily Montague (1769) by Emily Brooke, we see the familiar "American" format of Month/Day/Year; however, we also see the same format in British sources at least through the first half of the nineteenth century. Lord Byron (died 1824) dated all his letters in the "American" format, and his acquaintance Mary Shelley used the "American" format in the letters in Frankenstein (1818). Anne Brontë's The Tenant of Wildfell Hall (1848), an epistolary novel, also uses the "American" format. Wilkie Collins uses the "American" format in the Woman in White (1859); however, in the later work The Moonstone (1868), he uses the "American" format for the dates at the top of the letters but uses the "British" format when mentioning dates in the body of such letters. By 1897, we see in Bram Stoker's Dracula that the "British" format is used throughout Jonathan's diary.
Hi, everyone,
Welcome to AskHistorians! This is a place where people with questions about history can get answers from those with expert-level knowledge in the topic at hand. Thus, we require answers to be in-depth, comprehensive, and to reflect current academic scholarship.
Methods of both thinking of a date and writing it can absolutely be historicized--when Martin Luther sent his 95 these to the archbishop of Mainz, he dated it "Vigil of All Saints MDXvii." It was very much the convention at the time in Europe to date by liturgical calendar--it was a standard, and the standard eventually secularized.
Which is to say: no, saying "look at the Declaration of Independence" is not a sufficient answer for this thread. (Even without considering the look you should instead be giving to contemporary letters and a century's worth of news pamphlets). You'd want to think about prominent individual documents or types of documents (newspapers? guides to handwriting and letter-writing? a popular schoolbook? government handbills?) that might pick up one now-common date formatting before others, and how it could have spread.
While we wait for an answer, you might be interested in checking out some of the best answers in the subreddit, from our weekly Sunday Digest, our monthly Best Of competition, and our Twitter and Facebook accounts. We do, statistically, have a 99-100% answer rate for the top 50 questions each month. But due to the standards our readers have come to expect and treasure from AskHistorians, sometimes that can take a little time.
Please do not clutter the thread with posts that are not in-depth and comprehensive answers. It is not fair to OP. You will be banned.
Thanks for your patience!