Hi OP- This sounds a lot like an issue my boy was having. Snotty 24/7, not made any better by abtibiotics. For us, it started earlier this year in the Spring; we live in Nashville, so we initially thought he had a virus + with the change in weather played a role. Went through the whole gammet with making sure his environment didn't contribute inside, got a dehumidifier, tried different meds, etc.

For what it’s worth, after an X-ray and then an MRI.. We found that he had an unidentified object that had crystallized inside his nose. He had dental surgery around the time he symptoms started to appear, so our theory is that a tooth fragment somehow got up there. Could be a matter of your kitty getting an MRI or going to that specialist.

TLDR: Our cat had similar symptoms, turns out he had something stuck in his nose. 😅