Coughing and i feel like i have dust in my throat?

I have had this cough for about a month or two and what it is is that ill just be having a normal day and then ill just start coughing out of no where and it feels like i have dust in my throat ive even been drinking cough medicine and even with that it will not go away, then after a few minute cough attack it stops and wont happen again for a little while maybe an hour or two, and its so bad that i even wake up in the middle of the night to cough, i have cat who sleeps in my room im 22 and have no allergies to animals nor do i have asthma, i use to work at the post office and working there i was told i could possibly be exposed to asbestos maybe this has something to do with it?? but can someone please help me this is really really irritating