I suppose it was used for some transactions. After all, the pound, like the franc and the peso, are legitimate forms of "money." However, with the introduction of "American money," I would theorize that everyone wanted "our money" after independence was achieved.
Alexander Hamilton was the one who REALLY put the U.S. monetary system into place, though. That's why his picture is on the $20 bill. Here's more about that:
Hamilton proposed a national bank. Congress approved the idea in 1791. The bank had $10 million in capital. It could lend the government money and pay off state debts. Hamilton’s system also created a federal system to collect taxes.
But not everyone accepted Hamilton’s views. Many of President Washington’s advisors – called his cabinet – opposed Hamilton.
Opponents expressed many objections to Hamilton’s Bank of the United States. Generally, members of Congress from the northern states supported the idea, while those from southern states opposed it.
Another political leader, Thomas Jefferson, said the Bank exceeded the powers of the Constitution.
Hamilton defended the Bank. He argued for a broad interpretation of the Constitution. He thought it permitted the federal government to do what it needed to do to strengthen the country’s economic system.
Hamilton largely won his political arguments. He became the leader of nation’s first political party, called the Federalist Party.