Hi there,
I'm happy to help with your research on Einstein's move to America. After some digging, I was able to find that Einstein first moved to America in 1933. He left Germany due to the rise of the Nazi party and their anti-Semitic policies. He initially traveled to the United States for a visiting professorship at the California Institute of Technology (Caltech) in Pasadena. However, he decided to stay in the US permanently and became a professor at Princeton University in 1933 as well.
The letter that you mentioned, in which Einstein urged President Roosevelt to research the potential of atomic weapons, was written in 1939. This was six years after he had already settled in America. It's important to note that his move to America was not solely motivated by the atomic bomb research, but rather a combination of personal and political reasons.
I hope this information helps with your paper. Best of luck with your research!