Sorry to read about your difficulties. While you state you do not have allergies, have been tested, and seen numerous professionals, you do mention that you have asthma.
Asthma is the body's reaction to a sensitivity - typically an allergy. This is happening inside the lungs. The airways in the lungs cause the body to have increased air passage swelling (inclusive of nasal and throat), increase mucous production, and narrowing of airways (again nasal, throat, lung alveoli, etc.) This can cause air trapping in the lungs. Medication is usually needed to help decrease the swelling, open the passages of the lungs, and get rid of the mucous which is happening in the lungs and airways. There are many types of asthma - one which is considered Cough Variant asthma - which causes the coughing to clear the air passages - with or without wheezing - the sound which is heard when the airway passages have the above changes. Yes, runny nose causes the post-nasal drip, which goes down the throat, causing the asthma symptoms in the body. Asthma is considered an allergic reaction inside the body. Eczema is considered an allergic reaction outside the body - so eczema patients should always be on the look-out for asthma signs/symptoms.
It seems that you are on the right medications of steroid inhaler/nebulizer/spray. Antihistamines such as Zyrtec may be helpful as well. This decreases the histamine reaction which causes asthma as well. While not having "allergies" you are being treated for allergies for control of your asthma.
You can also look into Dextromethorphan (DXM). It is a common over-the-counter antitussive (cough suppressant) found in products like Robitussin and Delsym, used to temporarily treat coughs caused by colds or inhaled irritants. which is an active ingredient in cold/flu medication - but not without the go ahead from your provider. Sometimes Guaifenesin is a widely used over-the-counter expectorant (Mucinex, Robitussin) which can make the asthma worse. Any consideration of over the counter medication should be discussed with your provider.
A professional group which you may look into is called the Asthma Educators Association. They are a professional group consisting of Nurses, Respiratory Therapists, and Health care providers (NP, MD, etc.) who specialize in the treatments of asthma. They go through extensive training in the treatment of asthma - and take an exam called the the AE-C - to be Certified Asthma Educators. They may be able to shed some light in further information and treatment which could help you. Good luck and feel better.