Hello,
In my opinion you are explaining signs of an infection spreading and most likely related to the tooth. Your tooth broke and you say completely fell out, but a back tooth has multiple roots and part of the tooth may still be uner the gum. I would suggest you see a dentist, take an x-ray and most likely start antibiotic treatment to control the spread of infection. You would need to extract any remaining root under the gum. If you damaged the gum area, salt water rinses would help healing and I would not expect you to explain increased discomfort with your throat. Do you see an indent or hole in the gum? Do you feel anything hard, moving under the gum? This may be part if the toothacting like a splinter caught under the gum.
I would be happy to assist you further if you provide additional details or have more questions. Over the counter, anti-inflammatory medications may offer some relief until you can see a dentist.