Whenver I try to change the resolution in-game in the video options, the bottom half of my screen goes black. I can move my cursor in and out of the black part of the screen, but I can't interact with anything that's being covered with the blackness.

You can try to do it manually: http://pcgamingwiki.com/wiki/Medal_of_Honor#Widescreen_resolution
This way you will also be able to choose a custom resolution e.g. widescreen ones like 1336x768 or 1920x1080 ;)