It could very well be allergies then especially if it is seasonal. I went through a spring one year with a headache every day but no runny nose, sneezing, etc. My doctor said it was allergies. And I do notice that I get more headaches during allergy season both spring and fall.
I went to a Dr and she said it sounded like cluster headaches. That night I started getting dizzy and throwing up so my husband took me to the ER. They did a cat scan and said everything looked good. So I never did find out what it is. I don't think it's cluster headaches tho. It's not always on one side, some times it moves around and I always have a stuffed up head like it's sinus. Could it be sinus and the Dr can't see it? She checked my ears and eyes and said she didn't see any fluid in my sinus. I seem to have this every year at this time but not to this extent - not the burning feeling. I may go to another Dr. I couldn't get into my reg Dr because they were booked up so I went to a walk in clinic. I'm looking for a new Dr.
There are sinus problems and then there are sinus infections. Have you seen the doctor to see if what you have is indeed an infection that requires antibiotics?