Is the HD drive on your old PC showing signs of age, or is it the machine as a whole? If the drive itself appears to be in good health, I think that I would just buy one of those inexpensive external drive cases which you can plug into a USB hole on your new computer. This approaching doesn't require copying anything, though of course you probably can't execute any of the software on the old drive.
I've done that with the HD from my machine which suddenly completely died. Turned out there was nothing wrong with its hard drive.