Do you know how your MD ruled out asthma? Sometimes asthma can be difficult to diagnose, especially if it is atypical. I also will have an asthma attack when I laugh too hard. Some people will have one when they breathe cold air or exercise. Have you seen an allergist or a pulmonologist?
Since your MD thinks you have GERD, it isn't too far off the mark to suspect that you also may have asthma. GERD, post nasal drip, and asthma often go hand in hand. They all often trigger and/or aggravate each other. I have been diagnosed with and am being treated for all three. It was only when my pulmonologist started treating me for all three that I got my cough somewhat under control.
If the prilosec doesn't get the cough under control in a couple of weeks, I would either go back to my doctor or see an allergist or pulmonologist. In the meantime, elevating the head of your bed 6 inches might help with the GERD. I use inexpensive bed risers I bought at Walmart.
Also, it sounds like you may also have a post nasal drip problem. GERD can cause excessive throat clearing and cough, but so can post nasal drip. I use a first generation antihistamine, ChlorTabs. The second generation ones (Zyrtec/Allegra) don't work as well on PND cough. Its cheap and OTC and may be worth a try.