To have a unique domain structure, you should redirect www to nonwww using 301 redirect as other members said too. And regarding changing your webhosting, the best way is to migrate your website content at first and when everything was ready at your new hosting, then change nameservers of your domain through the control panel of your domain, In this case your website faces least downtime. Changing hosting only changes the IP of your website and search engines understand it and its a normal procedure for almost all websites to change their webhosting. So only try to do it with lowest possible downtime.
