Should I follow a specific drawing way? I usually enjoy doing things my own way, as it feels more natural. I recently started to take watercolor lessons(this was probably a mistake) and my teacher said my way of sketching was wrong.

For example, when I start to draw a face, I usually start from the hair, then I sketch the cheeks and the chin. Finally I place down eyes, nose, etc. I use very few searching lines.

My teacher says that the absolute correct way it to draw a circle, and place the face with a lot of searching lines.

My question is: Is there an absolute drawing style I should follow? Is my style outright wrong? I can get the positions of the thing I am trying to draw perfectly with my style. Should I still change it?
Depends. There is not one drawing style to follow. But if a learned technique helps you accomplish your task then, sure why not. That's the point of an education. An education teaches you how to use available tools to solve problems.

Math, Science, History, Literature, Philosophy and Art classes are all there to educate you about what had come before and how people have worked through those problems and come to a solution.

Education in different art styles and techniques is important because that knowledge empowers you by giving you the ability to solve problems you will encounter in your artistic pursuits.

When you are sitting in a class and learning a techique think of it as your are adding to your tool set. The tools are there to help you accomplish your goals.

Your goals can be accomplished by any number of application of techniques, tools and materials.

An education teaches you about the tools available and how others have solved the same, or similar, problems to what you have in front of you.

When making your art you will run into your own problems.

To finish that work of art you will need to come up with a solution.

An education shows you the tools and how to possibly solve your problem.

Your artistic goals are your own.

Your tools are there to serve you.

Not the other way around.