This is obviously a very difficult and frustrating problem for you. There are several important questions to be addressed.
1)Why do you get so many sinus infections? There are probably a variety of reasons possible for this. The most likely one is that the sinuses do not drain in a proper way and thus remain a source of infection even after weeks of antibiotics.
2)Which antibiotics will work? This is a very difficult question. There are several stronger antibiotics that will work for a while. However, after you have had several courses of these, they are not as
effectiveEffective strength cough syrup either. Without putting a needle directly into the sinus to get a culture, it will still be a lot of "trial and error" with the antibiotics.
3)Do your asthma medications make it worse? While this is possible, it is not as likely. There are some people who develop fungal sinus infections due to taking prednisone. However, this is not common.
I would suggest directly approaching the sinus situation as the first priority.
Good luck