Is my leaving my family's home to live on my own considered a sin, and does it affect my dutifulness to my parents, especially given that my mother hinted at this, and I am committed to helping her, calling her, visiting her, and sending her money?

There is no objection to you living independently away from your mother as long as she is satisfied. Continue to be dutiful to her through what you can, such as visiting, communicating, and spending money. Hasten to marry as much as possible, for it is among the good deeds that should be expedited.