Birth control uses hormones, and they can actually cause problems you may be quite unhappy with. You could get relief by taking magnesium 500 mg every morning and calcium with vitamin D every night. Then start taking ibuprofen about two weeks before you are expected to start. Take two pills every day for about 3 days. If you time it right, it can reduce the pain during your period quite significantly.
Maybe just mention you have debilitating cramps and you want to see a doctor to find out if there are any options to help. Then, they don't hear the words "birth control" and take it the wrong way. If you've complained or been noticeably suffering from them, I think they won't be surprised by you asking to see a doctor.