ever since we installed central air in our home, the entire house is damp and musty. Could this be the root of the problem, or is it something worse?

You did not give any details of the counstruction, age, insulation, climate or other details.

But the most common cause for this is that a too large of AC unit is installed.

That will quickly drop the air tempature, but not remove much moisture from the air.

That can run the relative humidity up.

Also if the condensate drainage is not handled properly it can add to the problems.