What is happening when my A/C operates in "dehumidify" mode?
The fan is not running at high speed to prevent icing, but for maximum cooling of the space. The cause of icing is that the system is undercharged with refrigerant or the throttling valve (expansion valve) is malfunctioning so that the low side pressure is so low that the temperature of the evaporator coil is below the freezing point of water.

I have always been told that an a/c system is supposed to be operated so that the internal temperature in the evaporator coil is 38 to 40 F. I have always supposed that this allows a safety factor in the case where the expansion valve closes too far.

My understanding of the dehumidify mode is that the cooling cycle is run continuously, but at a lower fan speed. The lower fan speed means less cold air is circulated, but the automatically controlled expansion valve opens to maintain the evaporator coils at the set temperature (of say 40 F). By running continuously at a lower flow rate all the water vapor is removed from the room but the room is not overcooled.

Our 26 year old Carrier is a base model 12 SEER which (at 42 kBTU/h or 3-1/2 ton) is undersized for our house in Dallas summer heat. On very hot days (~> 100 F) the compressor will run continuously for 6 to 8 hours in the middle of the day. The result is very dry air inside and I have to use lip balm in the middle of the summer or my lips get very chapped.

I have not tried to measure the humidity inside, but I may do that. We keep the thermostat at 74 or 75 F. I sit under a vent at the kitchen table and like to have cold air from the vent bathing me. This air would be very dry so perhaps this is why I get the chapped lips.