(the dress) https://www.mediafire.com/file/gs6k...+morph.wso/file

Your mesh doesn't have a UV map at all, so it has no texture.
My guess (if you did actually make UVs) is that they may have been lost when merging with the TS3 body pieces- the UVs need to have the same name in order to merge otherwise only one will export.

I suspect you tried joining a geom to an object? If yes, that doesn't work. Your UV will be a small dot in the bottom left of the UV editor. Join geom to geom, object to object.

The black you see in TSRW is from the doubled mesh in too close proximity. Simskin shader works differently from simhair shader. Easiest thing to do is remove inner mesh and close the bottom.

This is actually possible, the UV maps just need to be renamed to be identical before merging.
.obj UV is named UV_Map, .geom UV is named UV_0
In Object Data Properties > UV Maps these can simply be renamed to be the same and the two will combine without issues. Otherwise, unmatching UVs get deleted.

Objects have no bone assignments.

Whether that has any impact would depend on what stage in the project the meshes were combined. Looks like they combined with Nysha's template body, which is a perfectly valid way to add the body to .geom clothing even if it's an .obj because you would be doing the bone assignments afterwards.

I don't know what stage makes a difference. I know that if I join a geom to an object, the geom UV ends up a dot in the lower left corner of UV editor, and joining an object to geom, the object UV is the dot in the corner. Either is useless to me.

I do not know of Nysha's template. I have my own template folder of .wsos, objects, geoms, skins, shoe bonds, mannequins etc. Reminds me... I really need to clean it out!

Once any type of mesh in Blender, whatever format it happened to exist as a file ceases to be all that relevant. Just with variations on what data they bring in with them- like vertex groups or materials.
By the same merit, any type of geometry in Blender can be joined into one mesh that can then be exported out as whatever you wish. The dot UV will be from the map that got deleted unwrapping itself, which it won't do if the UVs were merged successfully. Very useful for converting meshes from any source, or pulling something to .obj to get the edges marked sharp or to edit it in other software.

Might work slightly differently between different versions of Blender, but that's how it is in 2.93