Birth control pills are known to affect in two ways, estrogen in it will lower pain threshold levels and hence your more susceptible for pain and secondly the same estrogen may also disrupt blood clot and cause dry socket as you have mentioned.
Next Steps
It would be good idea to seek an appointment after 22 days of cycle, most clinical research data indicate a spike in estrogen levels during this period of your cycle and hence all the complications as mentioned.
Health Tips
So if you schedule it after 22 days, since the estrogen levels are low it’s less likely you will experience increased pain and most importantly less likely to have dry socket complication.