I finally figured it out!

I was so excited to come home last week and see a package from Tim Holtz in the mail!  And look at the fabulous prize he sent: Stamper's Anonymous stamps, Distress Inks, and a mini Glossy Accents.

I had been watching some of his demos online and was trying to get the hang of distressing edges using the foam pads.  I've made a few very pathetic attempts over the past year and usually just ended up getting frustrated.  But this time I tried his advice to go very lightly and hold the tool at an angle.  Lo and behold - no square ink marks to be seen!