Hmm. I'm not sure how it's possible to keep the exact same dates. Ignoring that issue, perhaps if the nation quickly fell into disarray post-independence, or perhaps even before the fighting had ceased, so that several states actually secede from the new confederacy out of frustration, but return when a new constitution is being drafted.
With the end result that future generations see the Ratification of the Constitution as the moment the country began, and not at independence.
And/or, perhaps the war for independence is much shorter. It could have realistically ended around 1778. Suppose that happens, and in combination with the above, people see the writing of the constitution and unification of the Union as more significant to the formation of the country than independence day, although the latter would still be celebrated.