Shaw made his Broadway debut in 1929, when he played the murderer Wyndham Brandon in Patrick Hamilton's stage thriller, "Rope's End". In 1929, he married Margaret Delamere and lived with her in the Albany, an apartment complex off of Piccadilly in London. The two would eventually have a daughter together named Drusilla (born 1932). He returned to the works of William Shakespeare in 1931, playing Claudio in "Measure for Measure" at London's Fortune Playhouse. In 1932 he once again played Romeo at the Embassy Theatre. Other works around this period included productions of Ivor Novello's "Sunshine Sisters" in 1933, "Double Door" alongside actress Sybil Thorndike in 1934, J.M. Barrie's "A Kiss for Cinderella" in 1937, and Robert Morley's "Goodness, How Sad" in 1938.