The answer to Billington's title question is: Yes. They were both, at various times. They were not, in today's term, "woke," but I don't think you can blame the 1950s and 1960s on them.
I would say that R&H were as pretty much as thoroughly "woke" as it was possible for anyone to be at that time -- especially Hammerstein. You CANNOT take people out of their time and expect them to have behaved according to modern-day standards of wokeness (some of which are not universally agreed upon anyway). Not when their time and place was America in the first half of the 20th century.