Some systemic diseases affect the gums. In a young woman, this would most likely be lupus. Are areas of your mouth red, at times or do you get an occasional sore in your mouth, often?

The throbbing indicates some kind of inflammation. So don't suck on mints or lozenges for a while, either. Do you have any other symptoms? Do you have swelling of the lips or tongue?
Do you have any other stomach pain or cramping besides that vomiting with the birth control?

There is an inherited condition, called hereditary angioedema which can affect the mouth, tongue and gums. One subtype of angioedema is triggered from female hormone use. It is not common, but your reactions to the birth control seem severe compared to most other women.

If this continues after you follow Bryanna's advice, I'd see a doctor for some blood work.