If most of the required vertices moves are rather small, you could use the Integrate tool or the topology validation process to snap vertices together. If you do either use copies (or make backups), or apply them in an edit session, since they change the input layer. You may have to play around with the tolerance setting to see what "snaps" most of the vertices together without unacceptably changing shapes.
The same can be said for the Snap editing tool.
If the foregoing can get a lot of the work done but you still have some manual cleanup there is an align to shape tool on the Advanced Editing toolbar.
Another possible approach with gaps and overlaps larger than an acceptable tolerance would be to clean your layer, so that all overlaps and gaps become polygons, and then run an Eliminate selecting these polygons perhaps by size, but filled gaps could be selected by their lack of attributes), and then running the tool. There is no pure clean tool for shapefiles or geodatabase feature classes as there is for coverages, but if you have ArcGIS Advanced you could convert to lines and points and then build polygons from lines and points, or convert to a coverage. ETGeowizards has a polygon clean tool with the option to create points wherever there are overlaps, but I think it is one of the paid tools.
