You woke up because they didn’t maintain the right amount of anesthetic. This really shouldn’t happen. If they didn’t also give you Novocain, (which they probably didn’t) the reason it hurt was because it’s painful! If you also had Novocain the pain should be greatly reduced at least. But pain is relative and very personal.
There is NO WAY you are gonna get through that procedure without feeling pain after the surgery, that is just absurd. I am pretty good about pain (being big, mean, and ugly) but I would expect at least a day of not a lot of fun without some fairly potent pain relief.
A reason for light anesthetic can be anything from concerns over patient breathing to not wanting the trouble of cleaning up after a patient gets sick from the gas to poor practice methods.
Try the ice. If it is really a problem, then call the dentist or the answering service and tell him you need pain relief. Since I can be a bit “in your face” if it was not immediately forth-coming I might wonder aloud if letting a patient wake up during surgery is something my lawyer might be interested in taking up...“Does this happen often in your practice doctor?
Rockhead’s response reminds me of the old song. "Ain't too hard (to get along with somebody else’s problems)".
By the way if you like the idea of listening to music and like to sing-a-long but have a lot of swelling, I recommend going heavy on the Leonard Cohen. The other two choices have (in my opinion) exceptional voices, as a singer Cohen is a great songwriter if you get my drift. So you will probably still be able to harmonize with him, significant swelling notwithstanding.
On receiving the Canadian Singer of the Year Award Cohen is purported to have said: “Nobody was more surprised then me!”