Hi Redhot,
Before therapy I also did not feel ashamed by my body, but accepting who I am has made me much more acutely aware of how miserable being male makes me. The reason being that even though I always knew I wanted to be female, I wouldn't accept it, and thus most of the time I felt numb in regards to being male.
What my gender therapist told me, that helped me accept myself, was that as everyday as my experience seems to me, cis gender people do not go through it. Also, my therapist told me that no one can diagnose you as transgender, but after speaking with him, focusing on what I really felt, and accepting it, I slowly accepted what had been staring at me my entire life.
Now that I know you are already seeing a counselor, I would follow CarlyMcx's advice, and ask your counselor to refer you to a gender therapist. Also, since you come from a conservative family, you have difficulty communicating, and you don't know who you are yet, I would consider looking for a part time job, and asking your parents what you could do in order to earn some money. I worry that your father won't take what you say well, and then you will tell yourself, "I'm not actually transgender anyway," and then you won't receive the treatment you desperately need now.
In regards to your question, "I feel lot more like a cross dresser who wants a woman's body.. Is there a label for that?" I know when I was going through the stage you are going through, I was desperate for a straight answer, so I will say that I believe anyone who wants to be the opposite sex is not cis gender. In order to know where you fall on the gender spectrum though, will require you to clear a lot clutter from your mind first.