Hi Mike,
I looked at the skp file. Let me itemize the issues.
i) Face orientation. Try to make the faces oriented consistently. whitish is front bluish is back. It is not a show stopper (you can create the same textures for both front and back, but then the left-right sidednes of the texture becomes confusing. Please look at my attached file with the faces monochrome option to see what I did.
ii) For the model that you have, you should better make two textures as you noted. When you try to make a single spherical projected texture the distortions become too large. You can look at the attached skp file to see how I textured the two sides separately. And then note that I left window sides etc out, because they force the mapping to be distorted too much. For this shape unwrap method does not work well.
iii) The methodology. Tgi3D provides an initial mapping with a default texture (if the faces are not already textured). The next step is to edit the default texture to the desired texture, using an image editor. Since the image editor can not change the mapping your edits need to follow the mapping of the default texture. That is, say if the window is mapped to a particular rectangle in your texture image you need to place your window image inside that rectangle. In the attached skp file you will find two materials corresponding to two sides of the lander dome. The texture image you have provided of course does not have the correct placement. Now you have to basically cut paste rotate scale distort etc from your texture image into the texture images that tgi3d provided initially.
you can PM me for your further questions if you want.
regards
Dropship send2.skp