Hey Miss Jill, if it makes you feel any better, general anesthesia is different than dental/oral surgery anesthesia - I had horrible experiences with dental work (drugs like novacaine don't seem to affect me as strongly as they should) and with wisdom tooth removal, but my TOS anesthesia and surgery (almost 8 weeks ago now) went very smoothly.

Anesthesia is a science, and it is extremely safe and effective nowadays. Anesthesiologists use a combination of drugs to ensure that you are unconscious, unaware of pain, but breathing and circulating normally. No surgeon is going to make an incision without you being fully anesthetized (unlike oral surgery) because (aside from ethical implications) they need you completely still. :-)
A lot of the injectable drugs that are used are completely reversible, which means that if something unexpected did start to go wrong, they can inject you with the reversal and wake you right back up. Inhaled gas anesthesia exits your body through respiration within minutes after they take you off of it. If you are nervous about the anesthesia, ask to speak with the anesthesiologist prior to your procedure and have him/her explain what the plan is for you.

I had cervical ribs (well, I still do on the left side) and I am extremely glad I got the surgery done. I'm already much better off than I was prior to surgery. I wish I'd had it done years ago. And if my left side starts acting up, I won't hesitate to have surgery done to remove that one, too.

Good luck to you!