What should I do if I'm consistently wheezing when I exhale? I've been working around pet dander for over 1 month.

Get an inhaler. It sounds like you are having bronchospasm from being exposed to animal dander. An inhaler taken prior to work should be able to stop the wheezing. Your physician can easily prescribe this for you.
You have asthma. Asthma is, by definition, blockage of exhalation. Sounds like you're in the presence of a known allergen. Keep the dog out of your bedroom, start taking an antihistamine, and... Oh yeah... See a doctor. These things tend to get worse with time.