After thinking about it, I want to suggest a different direction for the divergence.
The Original Post suggests Washington dies early in the siege due to a freak accident. But what if he died later, and with a more heroic death? Suppose the siege goes as it does, but rather than seek terms of capitulation, Cornwallis instead tries to lead a desperate sally to break the Siege Lines, and manages to catch Washington's unit. Eventually the sally fails, but in the process Washington is cornered and killed. That could actually help Unite the American People, at least culturally but probably politically as well.
After all, in our Time Line Washington was many things; a War Hero, the "Greatest President Evar", the savior of the country, father of America, etc. In this Time Line, he is almost all of those things, but also Dies a Martyr, and if you think he was deified in our Time Line, imagine what's going to happen in this Time Line.*
You may actually get people to start joining the Continental Army en masse, to pay tribute to the Great Hero Washington's legacy, to see his dream carried through. Americans everywhere pay tribute to his Legacy, he who gave everything, even his life, to save Liberty.
And this isn't even getting into what could happen if he has, or is attributed, some profound, easily quoted Last Words, probably about passing the Torch of Liberty, long may it burn, united, in the dark or something similarly cheesy.
*Bonus Points if TTL Mormonism or equivalent literally deifies him Heavenly Kingdom style.
Okay, being a Mormon and known a bit about my religion, the way that'd work out is if the church was set up differently. Like having deification. Much like how the Catholics do Saint status. Now, the way OTL Mormonism views Washington is that he was a chosen spirit, specifically selected for the task of helping birth the nation in his capacity as General and statesman.
I must ask though, why does Mormonism get such a particular interest on these forums? It's not offensive or anything to me, I'm just very curious on why. Every where else I've been, it's one of the "best it be not spoken of" topics.