Well, the condensing unit is outside and the air handler with indoor coil is in the attic. All coils have a dran pan with a female fitting where you attach pvc pipe for a drain. If the drain opening is in the blower compartment it needs to be trapped (U trap). If the drain opening is with the coil after the blower it does not need to be trapped (but I trap them anyway).
All coils have a drain pan. Some furnaces also have a drip pan under the furnace that is piped into the pvc.
You say the unit shuts off.
Condensate pumps have a switch that opens when the condensate well is full and the pump won't pump. This burns out pumps.
If you have a gravity drain there'd be no reason for the unit to shut down.
If you have a condensate pump and it is wired into the Low Voltage wiring look at the pump. Take a bucket, pull the 3/8" tube and see if it pees all its water. If so the tube is clogged. Blow it out.Then remove the 4 screws that hoild the pump to its well (unplug and cut power to the AC first) and clean it out. Re-assemble and see how it goes. You can artificially fill the well with a bottle of water to watch it pump (restore power first).
If by some chance there is a water safety associated with the indoor coil and it is gravity open the panel (use 5/16 nut driver) and look into the drain pan. Fish around and make sure nothing is clogging the opening. Check the slope of the unit and pvc drain to the outside it should be 1/4" per foot. You can carefully pour water into the drain pan to check if the pvc is plugged. If so, blow it out.
Perhaps when they tell you a drain pan is not needed they are referring to the large pan under the air handler. If they are talking about the drain in the unit for the coil find a new Company. AC first removes water from the air... and it can be gallons every day. It needs to go somewhere. An other poster is right, it should pee outdoors but sometimes you'll find it goes where the sinks drain.
Anyhoo if you still have problems break the concepts down and make another run at it.