Wet feed won't flow in a gravity feeder and the wet feed that sticks in there will mold and spoil.
If you feed wet feed you need to feed it in bowls -- offering only as much as they will eat before it can spoil and cleaning the bowls between batches.
I really think that if you were to clean and dry your feeder, fill it with dry feed, hang it back up, and let them alone -- offering nothing else whatsover in any quantity -- you'd soon find that they were eating what was available.
They might gripe and complain because they're used to you reswarding that behavior by giving them candy and cake, but within 3 days they'd eat what they have available. No healthy animal will starve itself in the presence of food.
AFTER they learn to eat the healthy feed you've put into their feeder you could, if you wanted to, reintroduce treats in limited quantities. But only if you're willing to keep it limited and not let them train you to feed them on candy and cake again.