These two adjectives are used as pejoratives: cheesy meaning something cheap or cheaply made, and corny meaning trite or schmaltzy. (Schmaltz is chicken fat in Yiddish. Go figure.)
Apparently, cheesy originated in the mid-19th century and alluded to the smell of overripe cheese. Perhaps that cheese was sold at a bargain rate, hence the adjective.
Corny arose when mail-order seed catalogs appeared in the early 1900s, and to fill out the spaces, seed companies would scatter the pages with cartoons and jokes. I imagine these were not of the highest level of sophistication, and the adjective corny became attached to them and their ilk.