I use only gas logs for heat and moisture drips from the air handling vents in every room. How can I stop this?

Sounds like these non-vented logs? If so, they are not designed for whole house heating for 2 reasons, possible CO buildup and their primary waste product is water, Your humidity in the house is high enough to condense on the cold ductwork and this causes the water. Nothing you can do short of opening the FP vent or use central heat with gas logs as an additional heat source.

Mainly, Due to filter, indoor coil and drain pan dust blocked. Service your unit completely and check.