If she's in daycare than her constant colds are normal. My daughter had a was constantly sick and constant runny nose her entire infant-preschool years. Totally stunk. She could just jump from one cold to the next so it seems like one long dragging cough.
My daughter was a preemie and was put on a nebulizer at 2mos.old - currently uses it too. They won't label her asthma just call her a wheezer because of insurance reasons when she's an adult they don't want to "label" her. We used it when she was a baby when her chest would retrack, and when she wheezed and now she's 6.5 years old doesn't really wheeze anymore but when she has a horrible cough I will pull it out and neb her. I'm not really suppose to I don't think but I think it works.I think being a preemie that when she does get a cold or upper respitory it hits her harder than other kids. This winter I invested in some humidifiers and it's made a world of a difference in our house with coughing, colds, eczema you name it. We both have one in our bedrooms.
If your not getting anywhere with your doctor take her somewhere else.