Currently, at work, we are in the process of removing old, moldy shelves to replace them. I noticed that I have had tightness in the chest, coughing and a little difficulty with my breathing. Should I be concerned?

Yes. Take a look at the links on the left of this page: Related Links | Mold | Indoor Air Quality | Air | US EPA

At the very least, you should be wearing an N95 respirator mask and goggles.

Any vacuuming should be done with a vacuum with a HEPA filter.

Depending on how bad things are, the mold may need to be removed professionally.

Redistributing the spores is inviting trouble. If they find their way into the mucus membranes they will come alive.

This could also be an early indication for you that you could have Asthma or be alergic to mold.