Your downloaded shapefile already had a coordinate system defined in its *.prj file:
PROJCS["NAD_1983_StatePlane_Texas_South_Central_FIPS_4204_Feet",GEOGCS["GCS_North_American_1983",DATUM["D_North_American_1983",SPHEROID["GRS_1980",6378137.0,298.257222101]],PRIMEM["Greenwich",0.0],UNIT["Degree",0.0174532925199433]],PROJECTION["Lambert_Conformal_Conic"],PARAMETER["False_Easting",1968500.0],PARAMETER["False_Northing",13123333.33333333],PARAMETER["Central_Meridian",-99.0],PARAMETER["Standard_Parallel_1",28.38333333333333],PARAMETER["Standard_Parallel_2",30.28333333333334],PARAMETER["Latitude_Of_Origin",27.83333333333333],UNIT["Foot_US",0.3048006096012192]]
You appear to have used Define Projection to change how the coordinate values (which are in feet) get interpreted (as meters). This does NOT project the coordinates.
I recommend that you try the Project tool instead.