Sew a drawstring bag from two kitchen towels. When you wash your lettuce, you can put it in the bag and shake so that the towels absorb the moisture. The towel bag can be washed and recycled. Use this for washing grapes too!
I always use real towels when I wash my veggies. I loath using paper towels both for the money lost, and the waste of paper. I save paper towels for things like draining cooked bacon.