The way the esri shapewrite works is that the DestDataset_ESRISHAPE is a folder, and the name of the shapefile comes from the feature type. Either dynamically from the fme_feature_type or set explicitly on your feature type.

There area couple of way sto do that.

For instance you can use a scipted one like this

DestDataset_SHAPE =

$FME_MacroValues(Dir_out)\\\\$FME_MacroValues(File_Out)

Dir_out and File_Out are your published parameters.

Shapefiles fanout on the writer will create a new shapefile per fanout unique name, like a group by. If you need to create a new folder for each shapefile output you need to use fanout dataset in the advanced section.

If I understand your question correctly then change the feature type name in the writers options to your published parameter. That should name each shapefile accordingly.

The filename can be defined in the Feature Types Properties using a published parameter or an attribute. This image shows where to look: