To move LW to a new computer, move the entire incomplete folder, the preferences folder (specifically the downloads.dat file inside prefs folder.) And of course your saved/& shared folders where your already downloaded files are.

After reinstalling LW on new computer, close it. Then replace the new incomplete folder with the old one, and go to preferences folder & drop the downloads.dat file inside & replace the new one. But that must only be done whilst LW is closed.