All of the suggested softwares are available on the Library GIS workstations.
Everything being equal, we recommend Global Mapper as overall it's the quickest and easiest way to export multiple shapefiles to one CAD file while also clipping to the area of interest. QGIS also exports multiple shapfiles to one CAD (DXF only) file while clipping to the viewing area and is the only Mac-compatible GIS software here. ArcMap, ArcGIS Pro, and FME Quick Translator export multiple shapefiles to one CAD file but don't clip to the area of interest during the export process.
Common errors when opening the DWG file in AutoCAD
When viewing the file in AutoCAD or other CAD software for the first time, you may get an error that essentially informs you that the file was not created in CAD software. That is fine - you still want to open it. You may also need to zoom to the extent if there doesn't seem to be any data in the CAD window.