I added the following lines to the loop and that seems to be working.
Code:

;Do the stuff inside the For each--next loop for each file found. 
For Each $File in $Files
  If $File
    ;Split the time at the : character and then join the parts with no separation characters. 
        $Time = Join(Split(@TIME,':'),"")
            ;Copy the file and rename it during the copy process. 
	    Copy $File 'X:\$StringUsername\' + $Date + '-' + $Time + '.WMA'
		   If @ERROR <> "0"
			$Msg = MessageBox (" File copy Failed. Please try again." + 
			" If you continue to have trouble, please call the Helpdesk.","Error")	
			LogAction ("File " + $File + " was renamed to: " + $Date + '-' + $Time + '.WMA')
			Quit
		   EndIf 
LogAction ("File " + $File + " was renamed to: " + $Date + '-' + $Time + '.WMA')
 ;Sleep 1 second to get the $Time var different for each file.
 Sleep 1
  EndIf
Next

I was previously trying to write it like this, but I could not make it work. I don't know what the real difference is. I like the idea of deleting each file as it is copied since the way I have it now will create duplicates if one file fails to copy. Thanks
Code:

$error=@error
If $error <> 0
$Msg = MessageBox (" File copy Failed. Please try again." + 
" If you continue to have trouble, please call the Helpdesk.","Error")	
QUIT