Olive/vegetable/mineral oil are all pretty good at getting rid of adhesive, with the upside that there are no fumes or anything to harm the animals afterward. This is what I've been most successful with. I'd just use tons of it and a putty knife...and persistence.
WD-40 would also work the same way, but probably more likely to leave behind dangerous residue.
I've also used baking soda and water, and the baking soda is abrasive so it works but I don't know if it would damage the melamine.