Can HVAC spread mold?

Yes, if the system sits in a damp basement, the coil pan is wet, or a return pulls from a moldy joist bay. The clue is a smell that appears when the fan runs and fades when it does not. Wiping a supply grille does not fix a dirty cabinet.

Metal duct can often be cleaned. Moldy fiberglass liner and waterlogged flex usually get replaced. We will not sell you all-new ductwork if the cabinet and the humidity were the actual problem.