Hi Kim, I'm so sorry this happened to you. It's no fun when the treatment for one problem leads to another one.
I agree with your dentist, this is very rare. In fact, I have never seen this in my 26 years.
Antibiotics can take a day or two. But, don't wait if you are not seeing significant improvement. Go back to your dentist or see your physician.
Since your dentist has not treated this problem very much, you may want a second opinion. An ear nose and throat doctor would be a good choice since your sinuses are involved.
It's not unusual to be sore after a dental injection but there are very few true allergies to dental anesthetics most negative reactions to dental anesthetics can be attributed to the epinepherine that is found in many types of dental numbing medicine. Those reactions can be intense but go away quickly.
Be sure to find out what kind of anesthetic (numbing medicine) you were given and ask for something different next time.
I'm sure your dentist has made note of your reaction, but you should have this information in your records too so you can make any other future dentist aware of it.
Most importantly, don't delay having it checked again of it doesn't improve.
Hope this helps