Hi,
Welcome to ehealthforum,
It is not necessary that cysts will return soon after stopping with the birth control pills. Your normal fertility takes some time to return back after stopping with birth control use. If you are having functional ovary, then one ovary is enough to get pregnant. You should consider discussing your pregnancy plans with your treating gynecologist/obstetrician. Go ahead with further evaluation to confirm regular ovulation and hormonal levels (rule out any underlying issues that can lead to infertility). If any abnormality is detected, appropriate treatment would be started to help you get pregnant. If cysts get developed again, go ahead with finding the root cause for the problem and get it treated appropriately.
Take care.