In the spirit of this question (migration of royalty to North America), while not the US, Grand Duchess Olga Alexandrovna Romanova emigrated to Canada from Russia via Denmark after the Russian Revolution. She arrived in Copenhagen, Denmark in the spring of 1920, I believe on Good Friday, and stayed there until the Sovjets occupied the easternmost part of Denmark at the end of the Second World War, when she was accused of collaborating with a Danish bishop against the Sovjets. She, and other Russians who had fled to Denmark in the wake of the Revolution, began to worry for their safety, and in 1948, Olga and her family went to London, where they stayed in preparation for their final move to Canada. Leaving England in June (they boarded their ship in Liverpool), they first set foot in Canada in Halifax, and travelled to Toronto, where the Grad Duchess Olga spent the rest of her life, eventually dying at the age of 78 on 24 November 1960.

For a time, Jérôme Bonaparte lived in Baltimore, Maryland, but he returned to France (and was forced to abandon his wife and son from Baltimore). His grandson was Secretary of the Navy and Attorney-General for President Teddy Roosevelt, and founded what would later be named the FBI.

Ertugrul Osman was the grandson of Ottoman Emperor Abdul Hamid II, and he would have become sultan and caliph as Osman V had the Ottoman Empire not been dissolved and succeeded by the Republic of Turkey.

He moved to New York City in 1939, and lived in a two-bedroom walk-up apartment on the Upper East Side until his death in 2009.

How are you defining "immigrating"? The 20-year rule means I don't need to attempt to list princesses of still-ruling European royal families who have been living and working in New York recently.

Not as important on the royal front but of longer standing as US residents, the two children of Princess Lee Radziwiłł (Jackie Bouvier Kennedy's sister) have lived and worked in the US -- apparently without using their titles of Prince and Princess.