Upgrading on your current server and then moving is an option. After the upgrade was complete you'd need to dump out all of the mysql data and move that over along with the complete UBB.threads installation and then alter the config file to match the new paths.
I can say if it's looking for the old url still, then somewhere in your config the old url is still there. One option would be to download everything to your local computer and use something like ultraedit to do a search and see where it's coming from. I'm not familiar with the older versions at all so I don't know what config files were used at the time.
