What version of the S3 do you have, and are you running the stock or a custom ROM?
If you check the subforum for your carrier and device, you may find methods to update your rooted device. In the Sprint subforum, for example, I posted links to pre-rooted stock ROMs, and instructions for updating, for the most recent Sprint update. Another member, Skunkape, did the same thing for earlier Sprit ROMs. If you can't find anything here, you might want to check the forums for your carrier / device on xda-developers, as there's a pretty good chance you'll find something there.
If you're using a custom ROM, your best bet is to wait for your ROM builder to provide an update. If you're not using a custom ROM, and can't find a way to update your rooted device, your options are to unroot (if one method didn't work, search for another method--there are bound to be more than one), or to flash a custom ROM.
Edit: regarding frequent crashes. While I can't be sure, of course, I think application issues are a more likely cause than missed OS updates. Try going into recovery and clearing cache. See if that helps. If not, you might want to consider a factory reset. After the reset, though, don't simply restore everything. Start fresh, reinstalling only a few key apps that you must have, and see if your phone is stable. If so, start reinstalling apps just a couple at a time, and use the phone for a day or so. If the phone is still stable, install a couple of more apps. Repeat.