This is a common issue on the A3. The root cause is a seized damper, which causes the plastic hinge to crack. Removing the pin is quite difficult and you may end up causing more damage instead of fixing it.
If you find a replacement glove box, it is possible to take your chrome trims and put them on the new box.
If I were you, I'd get a new damper and try to glue the hinge before shelling out for a new box. You can then move the new damper to the replacement box if gluing does not work.