I would call or write Greer first of all. It is important that the button head fit properly and slide in all the way in, in the slit of the adapter. A burr, nick or bend on the button head can obstruct that. Also the tiny ball and spring in the button head can simply be stuck. The ball movement is little, just 1-2mm but can be checked by pushing on it with a needle. Cheaper reproduced nipples does not have a ball and spring fitted at all.
I can't see how the Greer upper side of the adapter is like, but if it is a screw there it should be removable and the plastic, felt or cork sealing ring replacable for a thicker one. Or maybe the screw can be turned down? The other fault can be that grease is not flowing into the cavity in the bearing as it should, the hole is obstructed, full of dirt or perhaps the bushing has turned a bit, offset the hole.
Here is a picture of Greer adapter and the upper side of an adapter that Thomas Fickau made some years ago. The head opening is opposite direction made for pushing against the head, necessary when using a modern grease gun. And some pictures of original grease guns, they can be found in different versions. The opening is made for hook on and pull on the grease gun.
