A little background! Please feel free to ignore it. We tried to change terminology in ArcGIS to make it more precise. "Projection" doesn't make sense if the data's in lat/lon, so we use geographic coordinate system (which includes a datum, angular unit, etc), projected coordinate system (includes a geographic coord sys, map projection, parameters, linear unit), and vertical coordinate system (must be defined in connection with a geographic/projected coord sys).
The Define Projection tool or a data's property page in ArcCatalog let you define its coordinate system. Both methods update the metadata only--it doesn't change the data's extent or coordinate values. The Projection Tool (hate the name) will create a copy of a dataset, in the target coordinate system. The source data must already have its coordinate system defined (and it must be correct!). Many, many people make the mistake of using the Define Projection tool when they want the Project tool.
I'm unable to reproduce your import problem. I'm trying it through the data's property page in ArcCatalog, XY coordinate system tab. All that happens is that the coordinate system definition is changed, nothing else about the shapefile.
Beyond the Albers data, you have some NAD83 data that you're trying to project in a different coordinate system? Are they in lat/lon, rather than a projected coordinate system?
I've attached a prj file for the NAD83 Albers definition. You'll need to rename the extension from .txt to .prj (I can't upload .prj) and place it in the coordinate systems, projected coordinate systems folder of the ArcGIS installation.