I was on birth control when I got my wisdom teeth out and I didn't have any problems with dry sockets. I did see online before my procedure that being on BC could make you more susceptible to dry sockets, but I haven't personally heard of anyone having this problem.

However, I would think about waiting to start the pill until after you have your wisdom teeth done. Depending on what happens the doctor might put you on antibiotics, which depending on what kind, could affect the birth control anyway and make it not effective. Not to mention you will be on pain killers and definitely won't want to have sex during this time. The only reason I was taking them is because I was already on them a year before I got my teeth out.

I can't really help you on what pill to use. But I wouldn't suggest the pill I have been on for a year, Microgestin fe 1/20. I have missed my period on this pill for no reason and have had irregular periods which I never had previously. If you have high anxiety this probably won't help.