Hello peteinny,
Yes, uninstall the third copy of Office from the computer before you dispose of it. If possible, put the computer back to out-of-the-box condition using the manufacturer's recovery procedure to help safeguard any personal information that was on the
computer during the time you used it.
When you get the new computer, just install Office normally. Chances are that it will install and activate without a hitch, but incase there is a problem with activation, choose the option to activate by telephone.