I contacted my Solidworks support company (I run a subscription for my real job) and they looked into the VRML export issue, and found some information that helped me out. Here's what you have to do:
1. When coloring your part, don't color solid bodies. Rather, color faces and features. This is important to remember, because naturally I wanted to color the body of the part itself, so that I wouldn't have to color in features later that were going to be my "default" color for the part.
2. Save part, and open in an assembly. Arrange the parts as desired in the assembly, and then save the assembly.
3. Save the assembly as a part. I know, I know, seems strange, but it worked for me. This saves the whole thing as a part file again, with the copies/duplicates of the original part also intact, in a new part file. Part files handle the VRML export better than assemblies, I think.
4. Open the new part file you have created, and export THAT as your VRML file.
5. Finally, don't have too many colors/features/patterns in your part. When things start getting complicated, the VRML will still not export correctly. Keep it simple.
And that's it! It worked for me. It's a bit of a hassle, because then you need 3 separate part files (the original 1x part, the assembly file, and the assembly saved as a part). This may only be an issue if you are trying to do what I was trying to do, which was pattern multiples of a small part and export as a VRML file, but I find it hard to believe I'm the only one who has tried to do this on Shapeways. If I just wanted to export ONE part with no duplicates, everything seemed to be working fine.
Putting this here in hopes that somebody will find it useful. I'm off to order some parts!