You do need to shorten your stitches to about 15 to the inch. Sew past the stitching line about 1/8'. Your cross stitching is supposed to lock the previous ones. I had this problem too at first till I shortened my stitches. It's harder if you have to unstitch tho.
If your stitches are pulling out when you remove the paper, try running a pin down the seam first. Then gently pull the piece on the diagonal.
I use the cheapest paper I can buy for my copier and it works fine.