Skipping Periods

I am 20 years old. I am a lesbian. March makes two years that I have been skipping every other period. I also didn't have a period in both November and December. I got my period in January however I'm almost two weeks late for February's period. I have been to the doctor and he wants to put me on birth control however I am skeptical of that due to my mom getting cervical cancer at age 27. For the past two years I have also found it extremely hard to lose weight. I eat mostly fruits, veggies and nuts, and occasionally chicken, hamburger and tuna. I workout four times a week, approximately 2 hours each time. I don't understand what is going on. Can anyone give me some advice?