Is it possible to fix a collapsing nose without changing the shape?

It is difficult to answer your question without an exam. If you did not get an answer from your initial consult, seek out a board certified plastic surgeon with extensive rhinoplasty experience to obtain a second opinion. There should be answer to correct your breathing problem if it is caused by mechanical obstruction in the nose.

Hi,
Your breathing proeblem can be fixed underdoing Septoplasty Operation,
and during Septoplasty nothing related to appaerance of your nose is changed.
You will still have the same nose without breathing issues.
Regards,

Hello and thanks for your question!
When it comes to nasal obstruction, there are several structural areas of the nose where the blockage can be occurring: the septum, the inferior turbinates (along the floor of the nose), the external nasal valve and the internal nasal valve. Based on your description of your symptoms and the septal exam findings of your former ENT, it sounds like you may have some internal (+/-) external nasal valve collapse. However, it is impossible to know for sure what is going on or what the right answer is regarding treatment options until you are evaluated in person by a facial plastic surgeon. 
If you do indeed have collapse of the nasal valve which is causing the obstruction, this is typically repaired using one or more cartilage grafts that help stent open the valve to improve airflow. Unfortunately, these grafts typically widen the middle third of the nose (at least a little bit - but sometimes significantly) even if the grafts are on the smaller side... after all, the point is to widen the nasal valve and keep it more patent. 
I would recommend you schedule a consultation appointment with a surgeon and undergo a formal evaluation so that they can better guide you as to treatment options and expected postoperative outcomes.  
Best of luck!