After updating a value in a custom drop down list, the text represented in the work item is not updated
This is working as designed. The values in the dropdown list are not stored in a relational format and are treated as a text pick-list. Because of this, the value stored with the individual work items is actually a text value and not a relational value.
In order to update work items, it is necessary to use bulk edit to set the text value to the updated value