Well I think that it’s transgendered when they have surgery (not sure) and tranny usually means transvestite which I thought meant wearing the clothing/makeup without surgery.
Generally, yes, she was writing under both her male name and her new name, S. E. Cupp for an online paper. One of the newspapers readers wrote complaining to her that lately she had ‘gotten’ to write columns and the ‘better writer’ (her but writing under her former male name) was therefore being displaced. S. E. Cupp then wrote to explain that the two writers were both her.
I’m not sure how you jump from “she wrote using a man’s name” to “she was born a man and changed to a woman”. I heard her say one time that she wrote with the name “S.E.” because she found that if people thought a man was the author her writings were taken more seriously than if written by “Sarah Elizabeth”.