Names were not something of significance to the vast majority of the gentry and even the general public during the era post-Revolutionary War era. I am a graduate student and I've gone through quite literally thousands of letters by the leading figures of the day, (including Washington, Adams, Jefferson, Morris, etc.) and I've actually never seen any reference to the idea of changing their names to distinguish themselves from their British ancestral past. (You can also view many of these letters here and they are even keyword searchable!)

I think that one of the largest areas of misunderstandings from this period is the idea that the colonists from the start hated Great Britain and wanted to throw off all the shackles that once linked them to the Empire. This is in fact, not true. And because this isn't true, a desire to purge any resemblances to the Empire wouldn't be a desire. The majority of the gentry did not want complete independence until the war had already started (even in 1775 and early 1776, we have journals and letters from folks like Charles Carroll of Maryland who were still hoping for reconciliation from Britain). By the end of the war and heading into the 1780s, a severe depression had hit America, and you had areas of the country, like in Pennsylvania, where up to 43% of their populations lost their land due to foreclosures (Bouton, Terry. Taming Democracy: "The People," the Founders, and the Troubled Ending of the American Revolution Oxford University Press; March 16, 2009). Even the gentry, like Washington were in huge masses of debt, check out an excerpt from Washington to his mother in February of 1787:

“I have now demands upon me for more than 500 pounds and three hundred and forty odd of which is due for the tax of 1786 ; and I know not where, or where I shall receive one shilling which ti pay it. In the last two years I made no crops. In the first I was obliged to buy Corn and this year have none to sell, and my wheat is so bad I cannot either eat it myself nor sell it to others, and Tobaca, I make none…”

This of course is only a small listing of the problems the new America faced, but I hope it can help explain why things, such as name changes, just weren't important to most Americans.