Do you have a history of asthma? White willow bark, which is used for pain and is what aspirin is based on, can sometimes cause asthma attacks. Here's a link that talks about it. You might just not be able to handle aspirin right now. How much have you been taking when it makes you feel as though you have asthma?
Of course, the reason that it might cause an asthma attack is what you would really want to find out. If salicylic acid clears out estrogen (and it clears out uric acid, too), your reaction might indicate that there's something in your lungs that needs to be cleared out. But of course you don't want to actually bring on an asthma attack.
Have you tried sodium bicarbonate? That might be a better supplement in your case. But I just had a thought that if you mix sodium bicarbonate and some crushed aspirin in some orange or lemon juice, the salicylic acid might turn into sodium salicylate, and not bother you quite so much.
Whoops, forgot the link!