Hey Chloe! Ok so here is what you will need to do:
Upgrade from trial to a paid subscription on your NEW design (congrats btw!)
Disconnect your domain from your OLD website (you will be able to access this so long as you continue paying the subscription)
Connect that same domain to your NEW website
Make sure to set up 301 URL redirects from your OLD site to your NEW website
Should you decide to go back to the old website, you would want to connect the domain to the old site again, then probably update your redirects. Let me know if you have other questions 🙂