Illustrator adding stroke to text with stroke and drop shadow

Add the shadow as an extra fill with a transform effect to displace it:

Group the text, then add the white stroke at group level:

How are you applying it? Use the appearance panel, make sure you add the white stroke below the brown one and it must be thicker than the brown one, as the stroke starts at the edge of the type, not where the brown one ends. So if the brown stroke is 3 points, for example, you might want the white one to be 5 points.