Did you wash the fabric first? The problem may be the fabric rather than the freezer paper. If the fabric has not been washed, there may be some kind of temporary finish on it from the manufacturer.
Alternatively, it's possible you got a bad batch of freezer paper. I have ironed freezer paper to both washed and unwashed fabric, and used several different rolls, and have never had the problem you are having.
Is it Reynolds freezer paper? That is the only brand I use.
Given the situation you have now, what I would do is use tiny dots of Elmer's white washable school glue in each corner and iron to set. Or, use a washable glue stick in the same way. This will definitely provide adhesion. To remove the paper later on, just use your finger dipped in water to dampen the corners, wait a minute or two to allow the glue to soften, and peel off the paper.