I wasn't super happy with how my GE fridge water dispenser worked so I made up this little gadget. Works good but can leave some drips, so the key is to push your glass into the dispenser for water, then slide it down towards the ground when it's full. That way the drips are captured by the cup.

I used double sided tape to fix it to my fridge and it's been holding well for ~6 months. Make sure you install it with the right orientation as there's a bit of a clearance issue (wavy side towards the shelf)