As per your question, I would like to mention that pranayama, no doubt, is good for health and relieving from asthma, but one should come directly at pranayama discipline until and unless he/she has strengthen lungs' stamina.
You also mentioned that you had the problem of sinusitis which might have been aggravated to asthmatic / breathing problems. In this connection, i would suggest you to please undergo SUTRA NETI and JALA NETI under the guidance of qualified yoga instructor, so that you could get relief to some extent in breathing by clearing nasal passage.
Though there are eight types of pranayamas a/c to Sri Patanjali sutras of yoga, hatha pradipika, but i do not consider relevant here to mention since pranayamas are to be done only under proper guidance of a experienced and qualified yoga therapist/ instructor, since real pranayamas consist of three phases: puraka (inhalation); kumbhaka (retention)and rechaka (exhalation). These too have to be followed in particular ratio, being many ratios like = 1:2:1; 1:4:2, etc. An asthma patient is not supposed to do such practices independently.
However, many preliminary pranayamas exercises have got developed to enable one get relief from breathing problems and other allied diseases. I mention here for your convenience and you can do easily:
Naadi Shodhan - without retention:
(i)first exhale simply without any over strain through left nostril -
blocking your right nostril by keeping thumb of your right hand lightly
and pressing on right nostril ;
(ii)inhale through left nostril keeping calmness of your mind and breath -
keeping the position of right hand as it is.
(iii) exhale simply without any over strain through right nostril -
unblocking the right nostril - blocking your left nostril by keeping two
mid fingers of your right hand lightly and pressing on left nostril.
(iv) inhale through right nostril keeping calmness of your mind and
breath - keeping the position of right hand as it is.
This completes one cycle of the modified naadi shodhana. You may do first
15 cycles on the first day. Then you may raise to 20, then 25 and then 30-35 as per your convenience. Kapalbhati also helps.
Apart from that, Gomukhasana is good for asthmatic problems.
Yogic pranayamas can treat asthma only if supplemented by some naturopathic treatments like hot-cold fomentation on the chest, diet therapy, autosuggestion/ meditation, etc.