I would take cheesy to mean of substandard quality, rather than sentimental or cliched. I think the metaphor is based on the texture of cheese and the fact that it doesn't always do quite what you would want when you are cutting it. I think corny derives from the rural aspects of it, the kind of unsophisticated demeanor of someone from the open prairie where much of the corn (in the USn sense) is grown. Why not wheaty? Well, we do talk of that sort of person as being a hayseed.