How can I get my AppleScript code to move files?
It appears the following line is returning an empty list:

every file of folder (choose folder) whose name extension is filetype


As an experiment, try a different suffix matching approach:

every file of folder (choose folder) whose name ends with ".wad"


In the Finder, make sure the .wad files are in fact showing their complete name. Do this with Get Info on one wad file in the Finder. It is unlikely but possible that the true file suffix is being hidden and thus not being matched.