Kennel cough still lingering at 4 weeks

I have a 5yr old golden doodle who started with reverse sneezing, then turned in to the hacking cough after 1 week. Took her to the vet and they said everything seemed normal, heart was good, no murmurs, trachea seemed normal and not collapsing. Vet brought up taking chest x-rays but I declined, they prescribed 300 mg of doxycycline for 1 week. It’s been a week now since she has been off the doxycycline script and she is still coughing regularly and hacking at least a few times daily. In addition her nose seems very stuffed up, sounds like she is snoring while sleeping.

As a vet what would be the next step in finding a diagnosis? Does she need another round of antibiotics? Should I move forward with chest x-rays? Could it be something other than kennel cough at this point?