It would be good to know more about the well. Do you not have a written record of it? Do you know how deep the sell is, what depth the pump was set, and what the static level was at one time?
A pump going bad does not manufacture air. One way a pump can introduce air into the system is if it pumps a little faster than the well can recover. The water level gets low enough that it forms a vortex like the swirl you see in a draining bathtub.
My guess is that the water level is low and the recovery not fast enough. Can you push a small poly pipe down the well to a depth where you can no longer blow air through it like a straw? Since water pressure is .43 PSI per foot and you most likely cannot blow more than a couple PSI, you can get a rough estimate of the depth. Do it first when the well had time to recover and then do it agoin when you suspect the level has fallen. Compare the difference.