You don't have to make a new shortcut. As soon as I install a BF game, I look in the game folder, copy the name of the game exe file, right click on the desktop shortcut that the game installed, select Properties, and then highlight the launchgame.bfg that BF always puts in the Target box, and paste the game exe filename over it. I've been doing this for over 5 years and it has never caused any problems. The only time the game manager runs is when I install a new game.
No you don't "have" to make a new shortcut instead of editing the old one, but it avoids accidentally deleting the name of the game exe when you meant to copy it. If you have shaky or uncoordinated hands, it's very easy to accidentally delete the name of the game exe while trying to copy it by clicking on it. The alternative is to copy it manually, but this can be tricky for games with long names with spaces, and you can accidentally remove a necessary quote mark.
Shuts down fine on both my WinXP and Win7 machines
So you've played a Big Fish game using the client, exited the game, then attempted to move the game folder to another location on your hard drive (not copy it, but move it so it's no longer at the original location). Seeing the client window disappear after exiting the game isn't proof enough. The client window may close on my computer, but I know the client is still running because I get the "file or folder in use" error when I try to move the game folder.
Your computers may very well close down the client properly after exiting a game, but my laptop has a Dell version of XP and who knows what version of Windows the OP (Original Poster) has. Maybe it's a name brand computer like Dell or HP with a slightly different version of Windows from retail versions. Or maybe it's 9 years old like mine and has accumulated subtle registry abnormalities. It's not necessarily worth reinstalling Windows for this type of problem, but I doubt Big Fish will find a fix that works for all computers.