I struggled with exercise asthma for as long as I can remember up until about a year ago (currently 28). I had in my head that I'd never be athletic for this reason. Last year I started heavily lifting weights with a trainer in order to correct some of the slouch I'd developed from programming and mentioned to my trainer that I would never be a runner because of the breathing issues.
I'm not sure if his response to that came from a place of wisdom or not, but he effectively told me to "quit making excuses and get after it" and started integrating runs into my workout. My lunch issues were always exacerbated in cold weather, and this was in November where the mornings were around 38f. It started with ~1km runs and I honestly thought this guy was killing me. I was bringing an inhaler in my truck and would use it between the "warm up" runs and doing weights because my lungs tight and on fire.
I was doing this routine 3 days a week that entire winter, and the length of the warmup run would range from 1-3km. I started noticing the pain less and less as the weeks went by. I actually started enjoying it, and began running on the days I wasn't going to the gym. I worked from 1 mile runs to consistently doing 4-5 5ks a week, with the occasional longer run. I still try to do 3-4 5ks a week to this day.
I hadn't really thought about it until this comment, which is probably what prompted this long-winded response. For me though, the solution was to just keep doing it. Your lungs get stronger.
This is not medical advice, and could even be wrong.
However, I had a friend with with the same problem and he did have asthma. The thing that helped him the most was dialling it back completely. Run 1km, does that make you cough? Run 0.5km. Once you get a distance where you don't cough, run it for a week or so, then increment by another 0.5km. If you start coughing, reduce it again.
After getting a flu some years ago, I had a similar problem ever since it cleared up. I have no idea if it is like yours, so it might be a longshot; mine is classified as "mild exercise-induced asthma". It is worst in cold weather.
I've also found that the inhaler helps only marginally, but it does help to start out each run on a much slower than normal pace, even walking (sort of giving the airways time to adjust), and also starting out wearing a scarf or big collar over the airways to trap moisture in the airstream. It's kind of annoying to do as my usual routine was a good warmup then just start right-off at full pace, but since then, I kind of pay for that.
Good luck, I hope this helps or or you can find something that does!
You might be opening up your lungs more than usual during running, and dirty or dry air can be irritating to pockets that aren't used to use. Start with short runs to build up your tolerance. Running on wet days can help -- the air is cleaner, wet, and less irritating. Good nose breathing will humidify and filter the air through your sinuses instead of passing straight to the lungs. Huffing steam through your nose in the shower before or after can calm the irritated tissue. If that doesn't work, reactine or a puffer can help. A light puff deep in the lungs _before_ the run can help stave it off.
I've had that sometimes during the cold seasons, and usually take a puff of salbutamol before setting out. I still get it sometimes after a hard run or a race.
Maybe try reducing the dose of running (duration x intensity), until you don't suffer so much, and then slowly build up if you can over weeks. You can walk & run, couch-to-5K style, if it helps.
This little-by-little approach applies to most things in running - from training your VO2max to coming back from an injury, so it may help you. The key is to understand that the timeframe of adaptations can be weeks to months, and not to rush things.
Ever tried running while on vacation? I grew up in new england and always found that I had a lot of exercise intolerance. However, I could always do better when I was on vacation out west. Now that I live in CO, I find my exercise tolerance to be way better. Anecdotally, I find that humidity is a real performance suck for me, as well as seasonal mold.
If you cough then you needs different method of breathing. I coughs when I breathing through mouth because inhale/exhale make my mouth and throath dry. Try use nose for both.