Cat has swollen nasal page?

My cat is about 13 years old and has never had any health problems. She is an indoor only cat. About 2 months ago she started to seem like she had a cold. I read that they typically go away on their own. But after about 2-3 weeks she was still sneezing and congested, and her left eye was always runny (clear, watery). I took her to the vet and they said she seemed fine other than for how long the cold like symptoms were occurring. So they gave me antibiotics for her for 10 days. Did that, and she seemed ok, but about a week after the antibiotics she started sneezing and hacking again. This time, the left side of her nose inside is completely swollen, like a growth is in there. She still eats normally and hasn't changed her habits. But the vet just put her on the same Clavimox antibiotics again, and after a week now they don't seem to be helping. Any ideas what this could be? (she was tested for kitty leukemia and HIV and both came back negative.)