Ubuntu has a tool for adjusting screen resolution and refresh rate, however, I couldn't find an option for selecting how many colors it should use. As by default, it chose a resolution higher than recommended by my Monitor's manufacturer, I am concerned that the colors could be too high as well, which might explain why I am having redraw artifacts when moving windows around quickly.

How can I set the number of colors to be displayed in X?

sudo gedit /etc/X11/xorg.conf

Scroll down to Section Screen. Mine's included for reference.

Section "Screen"
Identifier "Default Screen"
Device "NVIDIA Corporation NV34 [GeForce FX 5200]"
Monitor "Generic Monitor"
DefaultDepth 24

Following this are a set of "Display" subsections for each color depth the driver understands for your display. Change the "DefaultDepth" to the number you want. It probably defaulted to 24 bit depth and you may need 16 bit or even less. My monitor wasn't recognized in the hardware scan so this is where I added "1280x1024" as a GUI option in the "Display" subsections. A sample subsection for this purpose follows:

SubSection "Display"
Depth 16
Modes "1280x1024" "1024x768" "800x600" "720x400" "640x480"
EndSubSection