Much credit to you Celticgirl for your determination to pedal your way out of a sedentary life. I'm asthmatic and my allergist has me on
a Symbicort inhaler to alleviate symptoms. Here in coastal Rhode Island, it is quite flat so to get stronger on hills I have to go out of my way to ride them. Though the symbicort is a great help with breathing, the legs don't get the workout needed for good hill climbing. Nevertheless, hills here are not longer than one mile but I've come to manage that with moderate burning in the knees. Your shortness of breadth suggests Pulmonologist or Allergist should have a look at you. Your legs should give out before your breathing.