About jiwoon being a physician
1)i believe, a time period where only two genders were taken into account male/female i dont think studies regarding identifying the gender just by studying the people's face were invented at that time(i may be wrong too in this part)
2) Jiwoon being an acupuncturist, would have known that she is a woman if he had checked her pulse, but due to some reasons he didnt and he wouldnt dare to just to satisfy his curiosity.
3) I think it is the way Hwi carries herself, her voice, her walk that it would be hard for anyone to know that she is actually a woman. In manga, Hwi's feminine side is often shown like how she constantly ill and weak physically, she doesnt even drink alcohol that makes people suspect that she seems like a woman while in drama her character is made very strong that she can carry every single activity a male of that time can.
4) If you remember, jiwoon did suspect that she seems like the woman he saw in the forest but the fact that the crown prince is actually is next to impossible for people at that time. Even a word from his mouth and he would be executed. So him pondering over the thought that the Crown prince is a woman is next to impossible for me.
Let's see in future episodes how he would find out in next episodes. My theories are after they kiss on the lips he may find out or may be he hears a monologue of hwi at their secret place that she is dami.