Sounds like asthma symptoms. Does it feel like your chest is rasping or is your breathing wheezy? Get the GP to test your peak flow. You'll need to take a peak flow meter away with you and record morning and evening readings for a couple of weeks.
My asthma always threatens to cut in about this time of year. In the summer I'm free of symptoms but about now I start using my 'preventer' inhaler and with care I am not troubled too much by symptoms. I also run with a buff over my mouth in cold weather which is manageable except you can't do intensive sessions! It's also the reason why I don't ever race in winter.
Asthma 'attacks' can be triggered by sudden changes in air temperature or humidity so I am not sure that steaming is a very good idea?