Drake repeatedly refers to himself as "black." In the 1930s, that word was considered a pejorative. "Colored" or "Negro" were the preferred, acceptable terms. The writers appear to have addressed this error in Season 2 with Drake now consistently using "colored".
In the series, Mother McKeegan is telling Sister Alice about letters of support pouring in from all over the country, and reads one from a listener from "Chesterfield, Missouri." Chesterfield is a suburb of St. Louis that was mostly uninhabited wilderness and floodplain farms until the late 1970s, and was not incorporated as a city until 1988. There was no Chesterfield in January 1932.