As ireneo says, the salmeterol component of your Advair is probably what's giving you the heart-pounding/anxiety combo. As you have noted, the immediate effect of the bronchodilator, though unpleasant, is not long-lasting. If it's too disturbing, talk to your doc about bronchodilators with fewer cardiac effects.
The fluticasone component of Advair is a steroid, and steroids, since they affect cellular machinery, do NOT work instantly.