The OP called in pain. She was told to wait three weeks.
Several years ago, my 4 yr old DD had a mouth pain and our insurance told us only three people could take care of it. All were out when I called. Insurance expanded the net and found one person 60 miles away.
They wanted me to bounce her around in pain and wait a few more days, regardless. Or, I could easily wait a couple of weeks, right? Till one of the three locals was back?
I finagled a cell phone number and called to find that the person 60 miles away would be in my town the next day at a satellite office. I got DD in first thing.
And, yes, I would have paid out of pocket cash to help DD. But, we pay for insurance by golly, and we got it covered! (This was for an oral surgeon, not our regular dentist)
There is a difference between, "I am in PAIN and need help NOW" and "I would like to schedule an appointment."
Talk to the dentist and see if he knows the details. Then decide.