Do you have a model that allows you to change the temperature? If so, lower the heat. I prefer medium to medium low heat for grilled cheese. It gives time for the cheese to melt, and the slow cooking means the bread toasts through more without burning the surface touching the pan.
Here is a list of temperatures. I am not sure what the lowest setting on a FG is (assuming it is adjustable), but I would guess it is in the medium and up range.
If not, you may want to just use a pan. I love my Forman Grill (got it for xmas over a decade ago and use it often), but it is good for specific things. A great grilled cheese is not one of them. Also, you are cooking it on both sides at once. Normally you have n * 2 minutes to melt the cheese as the bread toasts; you only get n or somewhere around that when using the FG.