Cockroach Allergy?

So about a week ago, I moved into a small studio apartment that has cockroaches. I subsequently developed a cough and a sore throat. Of course, it could simply be a cold. But my paranoid personality lead me to look up the correlation between cockroaches and my symptoms, and I learned that cockroaches leave droppings that can aggravate asthma. Although I had the landlord fumigate the place, and I have not seen any of the bugs in a couple days now, there are places where I can still see lots of roach shit. I'm probably breathing this all in. I always thought roaches were disgusting, creepy-crawley critters, but I never knew that they could affect my health.