If your data doesn't have to be too accurate, you could try moving the shapefile features by dragging it to its proper location.
- Click the Edit tool .
- Click the feature or features you want to move. Hold down the Shift key while clicking features to select additional features.
- Click and drag the feature or features to the desired location.
As for the Spatial Adjustment Tool, I think it's the Transformation Tool you want. According to the docs:
Transformations move or shift data within a coordinate system. They
are often used to convert data from unknown digitizer or scanner units
to real-world coordinates. Transformations can also be used to convert
units within a coordinate system, such as converting feet to meters.
To convert data between coordinate systems, such as geographic to UTM,
you should project the data instead.
Under the Transformation Tools, I think you need the Affine transformation
More specifically, the translation option as the feature are just shifted to the east of their proper positions.
Here is the workflow from the ArcGIS docs
While each of the spatial adjustment functions is used for a different
purpose, the steps for setting up and performing an adjustment are
essentially the same:
- Start ArcMap.
- Create a new map or open an existing one.
- Add the data you want to edit to your map.
- Add the Editor toolbar to ArcMap.
- Add the Spatial Adjustment toolbar to ArcMap.
- Start your edit session.
- Choose the input data for the adjustment.
- Choose a spatial adjustment method.
- Create displacement links.
- Perform the adjustment.
- Stop your edit session and save your edits.
There is no need to save the map—all edits made to the database will
automatically be reflected the next time you open the map.