Photos would help. If the pan is full, then the coil is cooling the air below the dew point. However, a full pan is not impeding the systems ability to remove moisture. A major air leak would though. Conditioned air needs to remain within the interior. Further, the returns need to be removing conditioned air, not humid, warm air from outside. If the returns are in the ceiling, leaks there could draw in humid air.
There are two effective ways to clear a clogged drain line, depending upon how it's designed and the equipment you have on hand.
If there's a "T" in the line near the coil, you can use an air compressor or even an electric leaf blower. However, some of this air may shoot out into the pan itself, spilling the water.
The other method is to use a wet/dry vac on the output side. Slide the larger hose over the outlet tube and close the gap with your fist or cellophane kitchen wrap, duct tape, etc. and let the vac do the dirty work! It'll not only suck the line clean, remove any mold from the p-trap, but also suck all the water out of the drip tray.
Repeat as needed!