After a fresh start today, I've had some success.
I began at this thread: HowTo on dual monitors
Moved on to this thread: HowTo on TwinView
I followed the steps and added only the following lines to my xorg.conf file under the "Device" section:
Option "TwinView" "True"
Option "TwinViewOrientation" "RightOf"
Option "UseEdidFreqs" "True"
Option "MetaModes" "1440x900,1440x900; 1440x900"
Option "UseDisplayDevice" "DFP"
The native resolutions of my two 19" monitors is 1440x900, so that is the only resolution I specified. After a restart, I finally had success, except for another small problem. The leftmost screen does not line up with the monitor, and the edges of windows are cut off. There is a vertical black bar on the leftmost monitor along the left side. Adjusting the horizontal and vertical positioning on in the monitors' menus did not solve the problem, as they did not allow for enough adjustment to compensate. The rightmost monitor, however, worked fine.
I tried switching the monitors' connections on the video card (one plugs in to the VGA port, one into the DVI-VGA converter, which is plugged into the DVI port). The leftmost monitor still showed its vertical black bar, the right still worked fine. This meant that the problem was to do with the monitor, and not the TwinView setup.
After some fiddling around with the on-screen display menus, I discovered a "reset to factory defaults" option. I tried this, and problem solved. I have yet to test video on my TwinView setup, or fullscreen applications/games.
EDIT: TwinView does also work on Linux Mint.