“Schmigadoon!,” an adaptation of an Apple TV series that gently mocks big, brassy Broadway shows, danced away the best new musical Tony Award. The prize for the best new play at the Tony Awards went to “Liberation,” about a consciousness-raising women’s group in 1970s Ohio, which earlier this year also won the Pulitzer Prize for drama. “Ragtime,” a big, soaring musical that depicts an America being remade by immigration, racial violence, industrial wealth and political unrest, won the best musical revival on Sunday. John Lithgow won for best lead actor in a play for his role in “Giant.”