Couple of followup items:
Topolgy rules are a feature of the geodatabase. So you will need to move from shapefiles to feature classes.
Eventually you will need to combine all of the centerlines into a single feature class in order to create a network dataset for routing through your region.
However, as Russell suggested it will be a good idea to review the snapping (either manually via ArcMap or using the snap tool) on a County by County basis. I would start by examining two adjacent counties with a few overlaps manually and then start to build tools, scripts, validation checks as you go so you can repeat the process more quickly for the other counties and have the ability to repeat the process as new centerlines are published from the various entities.