It's more probable that the increase of blood circulation is causing your itchiness as well as your sneezing (loosening up the mucous or even from the increased blood supply).
One thing....one really never outgrows asthma...it usually returns as an adult.
If you're having symptoms at this time, it would be good for you to get thoroughly tested by a respirologist or whomever your doc sends you to...and a prescription for possibly a preventative inhaler or a rescue inhaler or something that you can take before you exercise.
Another option for now (before you get to the doctor) could be to start using one of those white formed masks just as you're starting to exercise. It'll help keep the carbon dioxide in your lungs and help keep the irritation and inflammation down. Maybe after your lungs adjust to the increased breathing you can remove the mask and you'll be fine for the rest of your exercising.
Hope this helps,
It's all a matter of perspective!
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