If the draft is coming directly from the air conditioner, then a thermometer might show that it is colder than the bulk of the air in the office. This is because the air coming directly out of the air conditioner is of necessity colder than the average room air when the air conditioner is providing cooling. Eventually that cold air mixes with the existing air and the temperature evens out and the average gets a little colder in the process. Getting some hard data with a thermometer sounds like a good idea in any case.
The solution is to point the stream coming from the air conditioner away from people. Every air conditioner I've seen has at least some louvers for this purpose. Point the cold stream up so that it mixes with the existing air near the ceiling, then the air falling down on the humans will be warmer after the mixing and the flow will be more gentle.
Wind also makes humans feel colder, even when the air is at the same temperature. This is what the wind chill factor you often hear in weather reports is all about. There is a standardized formula that takes into account air speed, humidity, and temperature to model how "cold" it will feel to a human. You could take some measurements and apply this to tell the facilities guy how cold it feels after these effects are considered.
Again though, regardless of what the wind chill measurements tell you, part of the solution should be to aim the direct outflow of the air conditioner away from people. You might also point out to the facilities guy that raising the thermostat temperature when cooling will save power and therefore money. If you feel more comfortable with it set to a higher temperature, then that is a win-win all around.