@MrE I’m not sure how much help this will be, but it’s worth a shot… By default, Cloudflare nameservers have a specific naming convention: ‘name.ns.cloudflare.com’. E.g. bob.ns.cloudflare.com, mary.ns.cloudflare.com, etc.
There’s a somewhat old blogpost that outlines the reasoning behind these naming conventions that I believe is still relevant, though it may not be useful to solving your specific problem.
You can view your assigned nameservers on the Cloudflare site, DNS page, below the ‘DNS Records’ section. If you don’t see anything there then you should probably contact Cloudflare support though their ticketing system: support (at) cloudflare (dot) com. If you do see your assigned nameservers, log into your registrar and change the records to just the assigned ‘name.ns.cloudflare.com’ ones. Doing this will set you to the ‘Full DNS’ option if that’s what you want.
Once that is done you may use Cloudflare’s DNS for your A, CNAME, etc records. I believe you may upload your zoneedit DNS BIND export into Cloudflare’s DNS Records section.
Since you asked, here’s an extremely basic example of a working A & CNAME record setup under the ‘FULL DNS’ option:
I hope some of this information was useful - I’m not too familiar with zoneedit unfortunately