I had issues when I first started using tappits, but when you get the knack for them they are fantastic. The Jem cutters are the same - a bit tricky at first. I have used both with fondant (although most often I would mix gum tragacanth into it to make them dry a little stronger), and they worked fine.
The trick is to roll the fondant/gumpaste very thinly on parchment, and let it sit for a few minutes so the surface is not so sticky. I also dust with powdered sugar if needed. Then with the tappits I press the cutter and wriggle it so the fondant moves freely on the parchment and to make sure the cut is really clean, then give it a good whack on the bench to free the letter. (Personally I think a better name for 'tappits' would be 'whackits') Gently guide the letter back into shape with a dry paintbrush and leave to dry. The Jem cutters work best if I wriggle the cutter on the fondant, making sure the letter stays stuck to the parchment - so you want the parchment along with the fondant to move over your work surface (if that makes any sense?) The Jem letters need to stay on the parchment when you lift the cutter, because you can't whack the cutter to release the letter like you would with the tappits. After removing the dough from around the letter, you can then leave the letters to dry in place, or gently move them to your drying surface with a small offset spatula.