If you have Visualize in your license, I think there may be some additional options for exporting to mesh formats, so you might pose the question in that forum as well.
Most applications don't support color in STL, including SolidWorks. There is a macro that will export to OBJ here Free Solidworks OBJ Exporter v2.0 It does save the solid color with in an .MTL file with the OBJ file, and it may save decals and materials as they were applied (see below). You might also look at Simlab Composer. There's a trial available, it reads SolidWorks files directly and can convert to mesh formats.
In general there is not a full transfer of textures and materials from CAD apps to meshes, so it may be necessary to recreate that in a mesh/polygonal modelling application.