@lmkarlov-Hi. If you’ve been to ENTs and had CT scans etc. which show nothing, it is most likely nerves. The trigeminal nerve which is the most power nerve in the body, supplies the sensation to the nose. All of the mechanics such as sneeze response, runny nose and temperature sensation just to name a few. If these nerves were disrupted, they can cause odd sensation upon movement and temperature change.
Nerves will never show on any type of testing. Nerves can really only be described. Perhaps your nerves are just regenerating in that area. Doctors are really not helpful when it comes to nerve/pain explanations.