3DWS files loading as white usually indicates that it cant find it's textures, if it couldnt find the model, it would not show anything at all, so I believe that it is finding the actual model file...
The problem will be that you need to copy the textures you used on the level, along with the level, to wherever you are copying it to.
And also, as Matty Halewood mentioned, the default(or at least most of em) textures that come with 3DWS are not in a format that DarkGDK can use. They are in an .STF format. There are a couple of things you can do to get around that, depending on your level of programming skill...
I think, if you look around the leadwerks forums you can find a document that lists the file structure of the STF format, it would be quite simple to write your own loader to read the image info into a data structure of some kind, and transfer it to a memblock or straight to a GDK image(to save out again - ala converter) or whatever.
But, the easier way around the problem is to just NOT use the STF textures at all... go through the 3DWS folders and move them all/delete them so you dont need to worry about them. 3DWS supports alot of image formats, JPG, VTF(valve's tex format - yes you can use half-life 2/source textures if u want to - although GDK has no equivelant to the source texture-material system, you would need to write your own)and TGA just to name a few. So all you need to do is to track down/create a bunch of your own textures, in whichever format you choose to work with, and put them in the materials folder of 3DWS and select those textures only when texturing your levels.
A piece of advice though, each time you select a texture to use on your level, write it down somewhere... when it comes time to copy the level and texture to your project, its always handy to have an actual list of the used textures(for small levels this may seem pointless, but when you have a huge level, using hundreds of different textures, from possibly a list of thousands in your texture library, it quickly becomes a time saver to know whats been used and what hasnt)
And lastly, if you are seriously wanting to use 3DWS as your level designer(there is no reason not too, its a great piece of software) you could look into the .3DW(3D World Studio file format) loader for DarkGDK, it allows you to load your level directly from its .3DW file to GDK. It also has options to put together a lists of all the entities in your level and their properties(such as health pickups or ammo packs). So essentially, you can design your level completely and wholey inside 3DW, then just put code into place to use it inside GDK. There are a number of advantages to using the native 3DW format as opposed to a DBO or .X object.
Here is the thread with the loader for GDK :
http://forum.thegamecreators.com/?m=forum_view&t=125713&b=13&p=0
That loader was written by Pixel Perfect, based on DBPro 3DW loader code by Lost in Thought. As far as I know, the LIT loader has had some updates for the DBP 7.3 update, but its only for DBPro. I dont think it would be too difficult to translate LIT's newer code into its GDK equiv if you wanted to though.
If it ain't broke.... DONT FIX IT !!!