I think you can get a free subdomain and dynamic DNS service at desec.io, with this you should be able to keep the domain updated with your IP and point it at your home server. But you need to have a public IP from your ISP and not to be behind NAT.
Anither option is to use a Tailscale Funnel. You will have to use a .ts.net subdomain with them and they terminate TLS and re-encrypt for you. On the other hand it’s completely free, you get NAT traversal, an encrypted tunnel, and you don’t have to maintain the IP even if it’s dynamic.