I haven't had problems with freezer paper coming off like that. Perhaps others can give you some advice on that.
I did want to add that when doing signature blocks I have used a template made out of poster board or card stock that the quilt block fit into which allowed for seam allowances. If you just give someone a piece of fabric to write on some will go all the way to the edge and their name will be cut off. The template also gave a firm surface to write on. To make the template I used a solid piece of stock to make the back and the front I cut out the center so it was like a picture frame. They had to write within the cut out space. I taped one edge together like a hinge so you could easily put the fabric square in and take it out. If you are going to have someone working at the table where this is being done they could do that. Three to four frames is usually plenty as they are reused. I used Pigma pens so they didn't wash out.