Hi Al The following are two rules of physical science to keep in mind: 1. Warm air can hold more moisture than cool air 2. Moisture condenses on the coolest objects first
In your case, it sounds like a stagnent air problem. That is, warmer air with lots of moisture is getting trapped up in the cathedral ceiling, but is not diffusing out through the roof or gable ends due to moisture barriers (such as tar paper on the roof). As the room cools, gradually the woodwork cools enough to be the focal point of condensation for the moisture supersaturated air. What may be compounding this is moisture rising from the ground under your house or excess moisture generated in your house (hot showers, cooking, etc).
The fix must involve circulating the air trapped up in your cathedral ceiling. This can be done with ceiling fan(s) or by having an HVAC cold return installed up high, although this will probably be a bit expensive. If your A/C is working, it should be acting as a dehumidifier, although running a free-standing dehumidifier migh help.