Since this is all brand new, I will take a guess and say the installer probly forgot to knock the plug out of the side of the garbage disposal before connecting the hose to it. So there is some kind of clog, wether it be the airgap itself or between it and the connection to the garbage disposal.
The amount you turn on your water valve should have nothing to do with how fast it drains only how fast it fills up. There is a pump built into the dishwasher to discharge the dirty water.
EDIT: If the problem is that the plug from the disposal wasn't removed... just disconnect the hose (use something to catch the water) and insert a skinny flat screw driver into the barbed nipple and tap around the edges to break the plastic plug out. Be careful as not to get too rough and create drainage leaks. Oh,, unplug the electric to the disposal first, and when finished reach inside to remove the plastic plug or it could jam the disposal.