What brand of bags are you using? I never had any problems. I do like the ice cube tray idea though. I used it for the baby food I made, I never thought about using it for breast milk..
I bought milk storage bags. I never had a problem with any freezer burn. I have heard, not from experience that zip lock bags are not good for breast milk storage. Something anlut how sterilized the bags are? But that is just what I have heard.
I used to freeze mine in ice cube trays (each cube is one ounce) and then put the cubes in a Ziploc bag with the date. It was never freezer burnt and the cubes make it really easy.