I (correctly) georeferenced a JPEG to an existing shapefile that was in the correct coordinate system.
I then proceeded to create line feature classes by manually tracing some of the features (roads) from the JPEG.
The mistake: When creating the line feature classes, I used the wrong coordinate system for the feature datasets, and so the line feature classes are all in the wrong coordinate system.
The result is that the feature classes are appearing in the correct location (because I manually traced them over the georeferenced JPG) but are in the incorrect coordinate system. Essentially I need a way to update the coordinate system to the correct projection without changing the location of the lines.