If the ducts are located within a conditioned space, even if they are enclosed in a soffit, they should not be sweating. It sounds as though you are either, A) not conditioning that space (in which case the ducts will need to be insulated) or, B) there is excessive infiltration of unconditioned air getting into the soffit. In the latter case you would need to plug up or insulate the cause of the excessive infiltration.
Additionally get yourself a dehumidified to remove the humidity from the air.
I agree with @Max Tanenbaum ,
Additionally, make sure you have given the open soffit sufficient time to dry out ie: inside of drywall if not to wet and framing for the soffit.
I would also look for mold in the nooks and crannies.
It wasn't the HVAC sweating. That wouldn't happen in heat mode, only in AC mode (summer). But we found the problem. There is a drip leak in some plumbing from the bathroom above. The drips are landing on the ductwork which is causing the drywall damage in the basement below.
Thanks for your help!
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