Nick, right now condensation sounds to be the likely culprit, as you mentioned this began when you started using the furnace. I'm assuming you live in a part of the country where it is cold right now? Or, are you receiving rainfall? Can you reach the dormer window to see if the glass or sill is wet?
The beams you describe would be along the lines of what's called a "truss joist," and this could be part of the problem. If these members do not have large ports, or openings, in the uprights, then there is no airflow to the adjoining spaces, which means that every space needs to be vented at the eave. If they only put in vents every 4' or 5' at the eave, it's not doing the spaces in between any good, and you'll need to go to continuous soffit vent.
Vaulted ceilings are especially sensitive to insulation and ventilation needs; in part because inside heating inside wants to collect at the peak, and because of the limited space between the insulation and your roof decking.
I hate to be the bearer of bad news, but you were told right; if this is a condensation problem because the space was not insulated correctly, you may indeed need to remove the drywall on the ceilings to correct things. For the moment, let's hope that's not the case.
Post back with info on your location, weather, and any added thoughts you think might be relevant.