I found this expression in an English textbook "Shall I ask her to return your call"? I learned that "shall I ~?" is not that common in America and "should I" is used much more often in America .
Do you agree wth the information? Thank you.

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Yes. Shall is not used much in casual conversation. They don’t have exactly the same meaning though

Should do something doesn’t indicate that it will actually happen.

We should go to school. (But might not)
We shall go to school. (we will go to School)

Shall has a certainly or a promise implied to do something. In legal terms it means you are required to do it.
Person A shall reimburse Person B. Means that the first person A has to (is required to) reimburse the other person B.