That looks a lot like an Accuride slide or something similar. Such slides have mechanisms that let the sliding part come out of the part attached to the cabinet. That is without unscrewing anything. For one type of Accuride slide, here is a video:
Adjusting the face position of such a slide often involves moving the screws on the part attached to the cabinet. Some of the screw holes are slotted to help with such adjustment. Start by removing the sliding part from the cabinet so that the screws can be reached.