You added random Cloudflare nameservers to your domain before adding it to Cloudflare, that cannot work. Use the search with the error message you got and you will get plenty of options on how to fix that.
The main point is if I cannot register the domain in cloudflare, I can’t get the nameserver address. It shouldn’t affect I register a domain in cloudflare, right?
I purchased a new domain for a week already and it is able to search in who.is. Even the name servers haven’t fully propagated, it shouldn’t affect a user to register the domain in cloudflare.