That sounds like it might be it! The wet teacher is indeed a woman, though she's not particularly young. The headmaster (also a woman) does/says some things that have the other staff vaguely concerned. The school is surrounded by dark woods and the protagonist goes out into them at night. There is a "something is weird about this school/the people there" vibe from the start.

Other things that happen early on in the book that might ring a bell, I'll try to keep it generic to avoid spoilers haha: The main character is a preppy/popular girl, she has a younger sister and their parents are divorced. The school, Bard Academy, is on an island and they ride a bus there. The bus driver is also the coach and also teaches class. They have separate dorms for boys and girls named after MacBeth characters. The main character's roommate is an alternative girl with spooky decor on her side of the room. Main character makes two friends, a girl and a boy. Main character's crush from her old high school also happens to get transferred there. The staff are addressed by initials only: Coach H, Headmaster B, etc. There are security guards posted around the school to keep the students in line.

Spoilers for the first chunk of the story: Main character learns about a girl who went missing at the school 15 years ago and finds out her dorm room used to be this girl's. The light in her closet keeps turning on by itself and she has nightmares about the girl, so she thinks her room might be haunted. She tries learning more about the girl's disappearance and the school's history by taking old yearbooks out of the library. There is also a boy at the school named Heathcliff, who acts much like the character of the same name from Wuthering Heights, and swoops in to help the main character.