Do I have to go to a gynecologist to get birth control prescribed by my doctor?

No, go and see your normal doctor.  Tell him about your issues.  He will probably prescribe a triphasic hormonal birth control.  "Triphasil" is one of them.  Once your hormones are under control, your life might be easier.   Lastly, if the first one, after 3 months does not help, then request a different hormone ratio.  Maybe a biphasic one like L'Ovral.  Getting the right combination is best to help in these situations.  If you have acne, there are hormonal birth control methods that can help with that too.  And all of this should be covered by insurance.  

If you don't want to see your doctor, you could also go to a planned parenthood office.  They can also help you get the medicines that help with your issues.  They also do regular health screening at an affordable price.  

Nah.  Your family doctor or a sexual health clinic like Planned Parenthood are perfectly able to prescribe them.

Now as far as what I suspect your actual concern is, which is having somebody examine your vulva and vagina?  That might be slightly more problematic.  Doctors are now encouraged NOT to require a pelvic exam prior to writing a birth control prescription, but there's some older-school types who still do mandate it.  Planned Parenthood doesn't, my doctor doesn't, and you can always ask the doc beforehand.

Your regular doc can do it.