How do I load multiple shapefiles into the same existing feature class?

I have a hundreds of shapefiles and, based on the filename, I need to load it into one of three feature classes (points, polylines, or polygons). My code works until it tries to load the second file of a particular type. Then, I get an error “FC_NAME already exists”. Yes, it already exists, that’s exactly what I want. Now load the shapefile into the feature class, please!

How do I load a shapefile into an existing feature class?

Here’s the code, trimmed down to exclude logic that gets to this point:

result = arcpy.FeatureClassToFeatureClass_conversion(shp, outputLocation, outputPointsFC)