I bring it into ArcGIS Pro with its original projection. This projection shows it is in the correct location. I use the tool define projection to change it into North Dakota State Plane South coordinates and it ends up in another country--outside of its intended position.
Your projection is already defined as UTM NAD 83: This tool is intended for datasets that have an unknown or incorrect coordinate system defined.
You need to re-project the data into the what you want: https://pro.arcgis.com/en/pro-app/tool-reference/data-management/project.htm: Projects spatial data from one coordinate system to another.
A best practice if you want to change a projection on a feature class. First it is best to not define the projection for a Feature Class, you should define a projection for a Feature Dataset. If you want to change the projection you should create a new feature dataset with the projection you want. Then import the feature class into the new dataset.
If when you first add the data it is in the correct location (you've confirmed that!) then you should not use Define Projection. The project is already defined, and it is correct!
Use https://pro.arcgis.com/en/pro-app/tool-reference/data-management/project.htm and Project the data into your desired coordinate system.