If you do have surgery, it is really important that they remove JUST the cyst and leave your ovary. It is called cystectomy and not all gynecologists have the skills to do it. Removing even one ovary can permanently disrupt your body's hormones and cause all kinds of health problems as well as infertility. The other one may pick up the slack but there is no guarantee.
The ovaries and uterus are needed our whole lives to keep us healthy. The ovaries produce hormones our whole lives and the uterus and its ligaments (pelvic support structures) are needed for skeletal / structural integrity as well as keeping the bladder, bowel, and vagina where they belong.
Women's organs are removed much more often than necessary.
Best of luck to you.
Yes if I was you I would see a second doc asap. I can't take birth control either, the hormones in them are unnatural. I won't never take birth control pills again either. I would go for the surgery if I was you. Doctors always trying to get you to do things that don't agree with your body. Please listen to your body, it never lies to you. It alerts you when some thing isn't right. Good luck