There are generic versions of corticosteroids like Arnuity (fluticasone furoate powder) - some with exactly the same ingredient, others with similar ones like budesonide and beclomethasone.

Recent research has shown that an inhaled corticosteroid, which lessens the inflammation that is part of asthma, is more effective in slowing the progression of the disease than a bronchodilator like albuterol used alone. And the steroid is more effective in calming the cough.

I agree that some research for an affordable solution is a good idea for you. Over the years I have switched from brand-name Symbicort (about $600/month, $145 with my insurance) to Breyna (about $300/month, $75 with my insurance) to generic (about $230, $5 with my insurance).
Finally, don't overlook the possibility of a program like GoodRx to help cut the cost of medication.