The other answers don't appear to have addressed asthma, so this answer will focus on that.
The OP is in a good position because she has been diagnosed with asthma. Other athletes may have asthma or exercise-induced bronchoconstriction and not know it. I was once in this position, and I frequently didn't do that well in spring rides or races until I addressed the symptoms.
Diagnosing asthma or allergic rhinitis
Asthma's typical symptoms include wheezing, coughing, and shortness of breath. These symptoms often present after exercise. They can also present at rest, including at night or in the morning. More experienced athletes may have atypical symptoms, e.g. mainly cough or feeling out of shape.
Allergic rhinitis, the fancy term for hay fever, is a related condition. This is basically sensitivity to things like pollen. The symptoms are perhaps more obvious, and they include sneezing, snotty nose, and itchy eyes.
In doctors offices, asthma is often diagnosed by FEV testing (forced expiratory volume) both before and after administering a short-acting beta agonist like albuterol.
People in general should consider if they have a family history of asthma, or if they had trouble with skin or respiratory allergies as a child, or if you have trouble exercising in certain seasons (especially spring and fall). If any of these are true, please consider if you might fit some of the symptoms of asthma, and consider asking your primary care provider for more advice.
Generally, there are two classes of medication. Asthma will cause your airway to constrict. Reliever medication basically just reverses this, and these are typically short-acting beta agonists like albuterol/salbutamol. If you only have intermittent symptoms, a reliever medication alone may suffice. It appears acceptable to take a dose (usually two puffs) from an albuterol inhaler before exercise. The effect typically lasts 3-4 hours.
If this is not sufficient to control your symptoms and you need to use your reliever inhaler 3-4 times a day or more, or if you need to use your reliever inhaler outside of exercise (e.g. to relieve symptoms at night), you should discuss with your physician. They will likely prescribe a controller medication, like inhaled corticosteroids alone or in combination with a long-acting beta agonist.
Allergic rhinitis frequently causes post-nasal drip. This is a fancy term for snot dripping down your nose into the back of your throat, which itself inflames your airways. This can be controlled with antihistamine medication (e.g. Allegra), corticoid nasal spray, and eye drops.
Patients should note that inhaler technique is important. Many people fail to use the proper technique, which means they don't get the intended dose of the medication. Check in with your physician or the re-read instructions on the inhaler. Other asthma controller medications are available beyond these classes of drugs as well, but you will need to monitor your symptoms and discuss if your asthma is not controlled.
One of the cited articles emphasizes that a gradual warm-up routine is helpful to prevent symptoms. On the commute, you could simply taking the first few minutes at an easier pace. Pacing oneself is definitely a skill that many athletes have to learn.
Having your inhaler with you is important. It took me some time to learn to pack my inhaler for my rides. In my case, it was a bit more difficult because my symptoms are rather intermittent. I need to take my inhaler before cyclocross races in the fall, but it is not that important at other times of the year, and I would frequently forget to even pack my inhaler before a cyclocross race. If the OP's symptoms are more consistent, it may be easier for her to just pack her inhaler every time she rides.
Ultimately, I think that asthma and related conditions are controllable and will not interfere with most athletic goals. Learning to control them does take some additional cognitive effort, but it is worth it. I wish the OP the best.