Moving shortcuts between folders

Background:
God help me - I saw nothing wrong with WIN3.1's "program
groups" - so I make my own. I create a folder, name it,
say, "Games" and make it a shortcut on my desktop.
In "explorer" (which I still call 'File Manager' because
that makes more sense...) I've moved shortcuts to each
game into the Games folder. I open the Games shortcut on
the desktop and Ta-Daaaa -- there are all the shortcuts to
the games.

OK. So I got a new computer. The old one, with the bad
sector on the hard drive, had XP Home, as does the new
one. Now, when I make a shortcut and move it, it
becomes "invalid" - the path name is deleted. Attempting
to change properties, etc., results in a slap on the
knuckles. Trying to delete the now non-functioning
shortcut, I get "access is denied. Make sure file isn't
write-protected or in use..." Well, it's not.

MY BIG QUESTIONS: how the heck do I get these things to
function? How can I delete the ones I don't want and get
the ones I do to work where I want to put them?

I suppose I should mention I am the only user of this
computer and, as such, the administrator.

Thanks.

~~Ozee