"Benim yavaşım" is not a correct usage. You should say "Ben yavaşım" ( I am slow)
I am a doctor.
Ben bir doktorum. "-um" means "am" (Conjugation of the be in first-person singular)
Like "Sen bir doktorsun" (You are a doctor)
When it comes to "Benim doktorum", there is a possessive pronoun > my
However, if I just say "Doktorum" it may have two means: I am a doctor and my doctor.
Be sure that you can understand what it means in context.
For example, when I ask you "Who is he?" (bu kim?) As you know, we need a subject and it means "my doctor" (He is my doctor).
However, when I ask you "What do you do?" (Mesleğin ne?/Ne iş yapıyorsun?" You don't need a subject. Instead, you need an object. Then it means "I am a doctor". I hope I can explain it clearly.