Every Fall, Usually In September I Get A Chronic Cough.

Every fall, usually in September I get a chronic cough. I have a tickle with it that makes me cough. I feel like there's flem that won't go up or down. I cough about every 18 seconds, sometimes hard or long. My throat is sore from the coughing. A couple of times when it came on I had a low fever and chills. But not every time. I doctored with my family dr. for 8 years with it. They tried different things. The only thing that remotely helped was a steroid shot and z pac. Then that didn't help a few times. I told them to send me to a specialist. They sent me to ENT last year. He told me I had acid reflux. I was reluctant, why would I only get it in the fall? I followed his orders and took the meds anyway. After a while, I got better. But it's September, and it's back. Even though I'm on the acid reflux meds. I feel like I haven't found the diagnosis or treatment yet. I thought about an allergist, but I don't know if they would try any treatment without doing the allergy testing, and my insurance won't pay anything on that, and it's $2000.00. What does it sound like to you? And what type of Dr. should I try?