I cut some thick scrapbooking cardstock down to the right size and stuck it against the front of each drawer:
Yup, that's it. Didn't even tape it or anything. And look at the huge difference:
Things are looking so much more neat and tidy. I love it!
If you are worried about durability, I suppose you could laminate the cardstock. But then this project might take a whopping ten minutes, so you'll have to decide if you can spare the time. Ha ha.