The US Army had standardized dates as Month day, year by the War of 1812; I've recently been going through record books, and the clerks are consistent. Federal records published during the 1830s are consistently in this format. For example, see the records of Prize Money awarded for the Battle of Lake Erie, as published in American State Papers: Documents, Legislative and Executive, etc. series 23, 1834.
Older documents, and British records, are mostly Day-month-year, which makes the US Army and the Federal Government the innovator.
OTOH, Civil records are mostly DD-Month-YYYY "this day Detroit, 7th February, 1813.", and continue this way into the 1830's, where both systems are found in court records, receipts, and family bibles. You can see the change occuring in family registers.
By the late 1840s, at least in Michigan, most records, federal, local, and personal, have gone to the Month day, year format. You also see the decline and disappearance of the long s from the script.