Jon,
If you are using a shapefile as your source file and a feature class as your target it definitely is not the same workspace. From what I understand, you are creating this source shapefile from a .dwg file and are attempting to bring in the data into your target feature class.
Assuming what I have mentioned is correct, I would request you to confirm if both the source and target have same field name and data types and same geometry and projection system. If they are the same, you can simply use the load option from the context menu of the feature class to add the data from the shapefile to your feature class without even opening ArcMap.
I am sharing few help links which should help you out:
Loading data in the Catalog tree—ArcGIS Help | ArcGIS Desktop
In case this is not what you are trying please let me know.
Regards,
~Shan
~Shan