See a pulmonologist, lung specialist.
I loved to see this post, since I was just diagnosed with EIA. I am an RN and didn't even recognize the symptoms. My riding was generally much more work this spring/summer than previous years and I was even on the trainer this winter. I spent most of my rides coughing and short of breath, which only got better if I stopped for a 10 15 minute rest.
My pulmonologist gave me two inhaler, proventil and Foradil. Proventil is short acting for short rides and Foradil is for my long rides. I don't need maintanence meds since I only have issues with exertion, true exercise induce.
The pulmologist did tell me to include a good warm up since not warming up can make the asthma worse and cause an attack much sooner.
I just rode 60 miles on Sunday and could not believe the difference.
If I can't go fast, at least I look good.