When we turned our AC on for the first time this year, we noticed after a short while that water was pooling in the storage unit where the AC is located. We had it checked out by professionals and it's still a problem even though they cleaned out the pipes and checked the coil. Then today, I heard water dripping from inside the intake vent, opened it up and there was about 6 inches of water inside with a drip coming through the filter about where the coil would be. We are concerned that mold may become a problem because of this.

Sounds like your unit is either frezing up, or you have a bad drain pan. How old is this unit? And did the service tech check the freon?

It sounds like you have a plugged drain off your unit. Depending on the humidity where you live the unit can create a lot of water in the heat transfer so that is not unusual. The drain should be checked and if possible let some clorox solution run through it to help remove build up.