Yes, you are right, the water condenses from the air con unit. That is because the air has been reduced to a temperature below its saturation point, or in other words, the humidity of the air that leaves the air con is about 16 deg celsius or lower, with a relative humidity of 100%. So water condenses. After leaving the air con, the air is gradually heated up by the surrounding, and the relative humidity slowly drops. So if you happen to place your humidifier near the air con, you find that the moisture you tries to add into the room has condensed back into water before it travels very far.
So when you place the humidity sensor near the air con, you will be measuring a much higher RH (but of course you need to wait some time to see the change if you are using the analogue hygrometer).
Most of the dry cabinets do not have a feedback control. What they do have, is a humidistat that stretches when humidity is high and contracts when the humidity is low (much like a thermostat). Typically, the response of such devices are quite slow. About 30 min as what I have tested.