You can add the quick filter for with the new field by clicking in the upper right corner of the sheet that should be affected by the quick filter, select Quick Filters-->desired field. Hope this helps!
You are expecting the Dashboard to reflect the changes to a Worksheet, unfortunately a Worksheet and a Dashboard are separate concepts in Tableau. A Dashboard is a direct reflection of the Worksheets used at the time of Dashboard creation.
In order to reflect the changes of the Worksheet in the Dashboard you must first remove the Worksheet from the Dashboard then Add it back in. Presto, your filter should now be displayed in the Dashboard. Remember to re-upload your workbook to Tableau Public.
Hope this helps!