Drew's first appointment as a faculty instructor was for pathology at Howard University from 1935 to 1936. He then joined Freedman's Hospital, a federally operated facility associated with Howard University, as an instructor in surgery and an assistant surgeon. In 1938, Drew began graduate work at Columbia University in New York City on the award of a two-year Rockefeller fellowship in surgery. He then began postgraduate work, earning his Doctor of Science in Surgery at Columbia University. He spent time doing research at Columbia's Presbyterian Hospital and gave a doctoral thesis, "Banked Blood," based on an exhaustive study of blood preservation techniques. He earned a Doctor of Science in Medicine degree in 1940, becoming the first African American to do so. The District of Columbia chapter of the American Medical Association allowed only white doctors to join, consequently “... Drew died without ever being accepted for membership in the AMA.”