allergic to dust w. itchy throat and red sore nose

The symptoms you report (itchy throat and nose for 4-5 years) in combination with your preexisting condition (asthma) and data about allergy to dust and animals make it likely that you could be experiencing a chronic inflammation of the nose and throat probably caused by allergy. You can request inhalation-corticosteroids to calm inflammation and you can also avoid all the substances that can cause allergy. In the meantime, you can chew/spray/take oral antiseptics and deodorants for treating the bad smell from the mouth.