This is a know issue on Windows XP, and a patch will be released soon, allowing Windows XP users to change their resolutions more conveniently. Until that happens, there is a possible workaround, which requires editing a configuration file.
Open this file: C:\Documents and Settings\[userame]\My Documents\Klei\DoNotStarve\settings.ini
Find the [graphics] section, which should look like this:
window_x = 251
window_y = 163
windowed_width = 500
windowed_height = 500
fullscreen = true
fullscreen_width = 1920
fullscreen_height = 1080
refresh_rate = 60
Make sure that you set fullscreen = true, like in the section in bold, shown above. Please keep in mind that you must enter the correct resolution and refresh rate, appropriate to your monitor.