1997 Ford Taurus AC causing rough idle

My 1997 Ford Taurus has surging and rough idle issue when the AC is on. It seems to have the same issue to a lesser extent with the heat on. If the AC/heat control is off the engine runs and idles smoothly. When the AC is on the idle becomes rough when at stop lights and there is a noticeable stutter at speed even on the highway at 65. If you turn off the AC/ventilation control to off the issue is gone and the car runs fine. With the AC/ventilation control on there is a cycling/clicking sound from the engine compartment like something turning on and off about every 3-5 seconds which seems to mirror the stuttering.
I have replaced the fuel filter, changed the oil and filter, changed the spark plugs and wires, inspected the air filter and put a couple bottles of fuel injector cleaner through the motor but the rough idle and stuttering continues. Any help would be greatly appreciated.