Did you explode the text component to make it part of the box? You would need to do that and then delete the faces behind the letters so the component/group shows as solid.
The 3D text has faces on both the front and the back. When you explode the text onto the surface of the box, you need to get rid of the faces at the back. I made a quick example and show those faces as selected.
One issue I have is that the when I create 3d text, I specify 2mm extrude, but it places a MASSIVE text. Then I have to rescale it to like 5% of the original size, which I think means the 2mm becomes like 0.1mm depth.
Is there a way to specify the dimensions of the original text?
Your model would be so much easier to work with if you took the time to thoroughly clean it up. This is a sample from the SKP file from the link you provided earlier.