You can’t have a widget without code. The code NORMALLY comes in the form of a plugin.
So what you want is either a theme or a plugin that creates a classified-ads management system that has a widget.
Hence our answers are plugins for the most part. Now, if you could list the specifics of what you think this widget should do, or what the ad system should do, we may be able to narrow it down, but you’re kind of asking for something specific without giving us the details
(I am a WP expert and LakeRat, you really should consider yourself at least a WP Rockstar, cause you know a lot more than you think )