Clarification: I don’t have any Cloudflare experience. There is an Edge certificate in Cloudflare applied to the website. We are moving the website and need to change the IP addresses. Will this SSL automatically apply to the website with the new IPs?
Changing the IP address won’t affect the edge certificate (assuming the record remains proxied so it is used).
Make sure you have a suitable SSL certificate on your new origin and you are using the Full (strict) Cloudflare SSL mode so your connection is fully secured.