50 feet is a datum issue.
To mitigate the apparent non-moving issue, you should add data to a map with no other data because the first layer that is added to a map sets its coordinate system. Add other data after.
If your data doesn't have a defined coordinate system in the first place or one you are unsure of, then that is an important step. Examine the coordinate system via its properties.
Project should only be used once a shapefile has a defined coordinate system... a correct coordinate system, since defining something wrong will only make matters worse.
After using the Project tool, you can compare it to known/trusted data for placement... if it is off by 50 feet then it was defined wrong.
... sort of retired...