The first thing to come off of the anode is a molecule of AgOH which is dissolved in the water and ionizes to Ag+ +OH- which combines with another AgOH to make Ag2O + a water molecule. In doing so, an electron was added to the anode, allowing another electron to enter the cathode, which creates the current.
The surface of the anode has Ag+ ions, but they are not soluble by themselves, and neither is AgO- ion which would remain tightly bound to the anode. The AgO- ion could presumably bind with another Ag+ to make Ag2O which would enter into solution, but that would also deliver two electrons, one for each silver atom, keeping the current the same.