I stopped using sterile cotton balls at 4 months and changed to facial squares. Guardian sells 3 for $5 and sometimes Watson's has 1 for 1 deals. I use washcloth for her bath to clean her face but the facial squares are for after meals cos she tends to get rashes from the food and formula. I didn't mind the cost because it's definitely cleaner and softer for her delicate face.
I start using handkerchief when my boy is only 3 months. No harm trying but pls make sure you find the soft kind material.