You need to reorder the columns on the output feature type (writer) itself. Doing it in the AttributeManager will only reorder the order inside the workspace, not in the output.

Keep in mind, if you are writing to an existing geodatabase, you'll need to drop and create the table to alter the schema (see Data Loading, Updating and Deleting for more on Feature Type Handling Options).

If you click the triangle on the left of the writer feature type on the canvas, the order of the attributes should match what comes after the AttributeManager (unless you are using a manual attribute definition in the writer feature type).

If you are using an automatic attribute definition the attribute order in the writer feature type should match exactly as you have it in the AttributeManager.