Should I just build the new website on the new host, using a dummy domain name and then point the existing domain to the new host?
That would be the easiest option, yes - although you will need to review Changing_The_Site_URL for details of how to change the site's domain name when you are ready to launch the site.
Another option would be to develop the new site in a sub-folder of the existing domain and then see Giving WordPress Its Own Directory when you are ready to launch.
I prefer the latter approach myself. It's relatively simple and only results in about 5 minutes downtime compared to a max of 72 hours if you're waiting for DNS servers to update worldwide.