I have tried it again! Thank you Claire and Sarah for your advice. I put the stove on looooow heat and shredded the cheese and it turned out much better- not perfect, but better. And, turns out, I don't think I really like grilled cheeses. They kind of hurt my stomach for not much payoff. So I think I'm just going to let this one go; we'll call it a draw.
And yes, that is a giant jar of mayo in the background. I mean, come on. Why not?