Hi Rachael,
The points are in UTM zone 12 North, rather than 13 (very generally: Red Hill area of the county). Try defining the data as NAD 1983 UTM Zone 12N. At that point you can either let ArcMap project it on-the-fly to zone 13, or create a copy via ArcMap (right click in the table of contents, data, export data, choose to export in the data frame's coordinate system) or the Project Tool (once the points are in a shapefile or feature class).
Hope this helps,
Melita