Some games are just not going to support that resolution whatever you do as it's non-standard (Unreal Engine 3 games for example will either break or fail to apply some of the post processing or just plain run like total cr*p Bulletstorm for example).
Your best bet, if you don't want to buy a new monitor (and why should you!) is to use your internet search engine of choice to search for the games you want to play and 1680x1050 to see if other people have managed to get those games to work at that res in a way you can reproduce.
Some older games can be jiggered to work in almost any resolution, information on how to do this with most of them can be found here http://www.wsgf.org (they also have writeups on some newer games which may help you too).
Failing that (and if your monitor supports it or you can force it in drivers) try selecting a 16x9 resolution that will fit within the boundaries of your monitor since most games of the last few years (since about 2006) have been console ports and were ported from systems designed to display on 16x9 widescreen TVs rather than 16x10 monitors this may help a bit (although you'll get black borders or stretched images and you'll still get some upscaling artifacts they'll at least be better than 800x600 upscaled!)