Yes you can. Your Refined.Site exists in parallel with your regular Jira Service Desk portals. It isn't until you refer your customers to the new URL, or when you change the email templates, that the customer will end up at your new site.
With RefinedSites you get your very own subdomain running on refined.site. Setting this up basically means informing the entire internet (and its DNS-servers to update their caches) that there is a new page, and this might take some time. If you have been waiting longer than you think you should (although it might take in the lines of hours - the internet is big), you are welcome to let us know and we’ll make sure everything is running according to plan.
With RefinedSites we are running a separate service (in parallel) to regular Jira Service Desk, and as such the RefinedSites customer portals are running on our servers. This means the Customer Channels link in the agent view will link to a default JSD portal rather than a RefinedSites portal.