Dry sockets are miserable. I had one when my wisdom teeth were removed and wasn't able to sleep until I had it treated. It's unusual that your dentist isn't willing to help you with that. When we have a patient with a dry socket, we pack it for them at no charge even if it has to be done several times. I agree that it's not ideal to leave behind a root tip. I hope you get some relief soon.
I would strongly suggest seeing an oral surgeon to see if removing the root tip would be a good idea. Although it has never happened to me, my initial reaction is to attempt to remove the root tip or refer to an oral surgeon who can.
Do you believe that the pain that I'm in could be coming from the left behind root or from a dry socket? I've never had a dry socket and I've had all my wisdom teeth removed at the same time. I have never had problems like this and the pain just won't stop.
Strange. I would have recommended going to an oral surgeon and getting the root tip removed. I don't think leaving the root tip is an ideal solution. It can result in further sinus infections.