I've found a workaround. Bring the shapefile into ArcMap, export it to feature class using the layer's data source projection. It keeps the old projection (GCS). Open a new map, add a feature class with my preferred projection of State Plane Colorado North. Bring the newly exported feature class into ArcMap, export it using "The Data Frame" option for coordinate system in the export window.
Why my "Project" tool is not functioning properly is another issue, as you all were able to Project without issue. Happy to have my ducks/files in a row. Thanks for working through this with me.
Alex