Essentially, I have a dataset with drop down menus in one column, but I can't edit it or add new text.
I have managed to delete the original text classification in the field, but can't add 'Camel Race Track' to the DropDownMenu.
Your database has Domain or dataset has subtype. If you want to manage them you can add more parameters for the domain or subtype. see Create subtypes and attribute domains to manage domains :
to manage subtypes: