If I save a column set in a particular watchlist, then try and use that saved column set in another watchlist, the column set does not appear in the search.....why?

I have never seen the described behavior before. If I select Save Column, give the Column Set a name with Save as selected and select Save, it always shows up under My Columns when I select Change Columns.

Adding & Editing WatchList Columns
Saving & Re-Using Groups of Columns

Please contact technical support.

It happen: 'change list/scan properties', edit columns, add column

Select 'My Saved Columns': is empty

IF instead click on 'Change Colums' saved columns do appear in 'My Saved Columns'

I get exactly the same lists when following the two methods you describe to bring up the list of available WatchList Columns and Column Sets.