Sorry I haven't responded...I was out of the office last week.
So, I'm running thru the steps I think you ran thru using 10.5.1. In ArcCatalog, right-click on a folder (make sure it is not a fgdb folder) and select New->Shapefile I was able to add City and Statge and change the length as I added them
you could also do use the Add Field tool
And if you wanted to get fancied and more consistent, you could write a simple script and custom tool that prompted them for the new Shapefile name, and output folder (so they could write it to their own custom folder for example) and have it set up the whole things...then you could show them the script to teach them simple programming. just a thought.
As mentioned in my first post, you can grab a copy of the code from the Results tab (open with Geoprocessing->Results). Expand the Current Session (or previous, depending when you ran it), right click and copy as python snippet.
arcpy.AddField_management(in_table="D:/_temp/testAddMoreFields.shp", field_name="statecode", field_type="TEXT", field_precision="", field_scale="", field_length="2", field_alias="", field_is_nullable="NULLABLE", field_is_required="NON_REQUIRED", field_domain="")
Anyway, just some suggestions. But if you are sure you are copying the shapefile into a good folder (not a file gdb folder) and you are altering the length as you ADD the field (you may not be able to alter once it is created...in that case drop and add is better option) and you are still having issues, zip up the shapefile and I (or someone else here) can take a quick look....or contact tech support if you have maintenance and are a contact for your org.