As long as you have done the pulmonary function tests and all came back normal this is only a small chance that you have asthma.
However I would ask them how they decide if you have asthma or not, most of the values are based on averages so if you are "borderline" asthmatic, then there is a possibility that you have above average lung capacity and mild asthma.
Acid reflux is defintely a possibility of cause though so I would give the Aciphex a bit more time to work. There may be other drugs that work better for you, but you have to ask. (I know almost nothing about acid reflux though)
I think they test for asthma using pre and post bronchodilator performance on breathing tests. If this was clear then it is probably not asthma.... though of course it could still be asthma if it is exacerbated by a certain trigger.