Why would a sprinkler Air Valve leak after winterization?
Without a picture I am guessing, but it sounds like the "something connected above the air valve" is your pressure vacuum breaker.  A pressure vacuum breaker prevents water from flowing from the outside back into your main water line (and then into your drinking water).  It will allow water from inside your water main to travel through it with the water pressure from your house forcing open the pressure vacuum breaker (while water traveling in the reverse direction closes the valve and prevents contamination of your drinking water).  

It does sound like the sprinkler water main has a very slow leak.  And I imagine a slow leak from your sprinkler water main is not enough to completely open the pressure vacuum breaker and it is just exiting at this point (when the air valve is closed).