@Charl Grové , Firstly, apologies for any inconvenience with this issue. I understand you have referred to multiple articles and tried multiple steps to resolve the issue.
You have mentioned that you contacted Microsoft support, do you have a #SR ticket number? Could you please share so I can track to fetch additional info?
Just to clarify, I believe you’re referring App Service domain which was bought as mentioned in this doc – yes, App Service Domains use GoDaddy for domain registration and Azure DNS to host the domains
https://learn.microsoft.com/azure/app-service/manage-custom-dns-buy-domain
You have mentioned that, “I have domain within one of my Azure subscriptions” and “I’ve done this, however when in the portal it says I have no domains". Could you please confirm you have logged into the right subscription directory? and then, “Is your domain listed under the App Service Domains blade and on DNS Zones?
Transferring of domain out of Azure (not a registrar) to another registrar (in your case Cloudflare) is supported - you may try via API and PowerShell. You will need to get an authorization code which you can take to the registrar of choice to proceed with transferring out of your domain.
Please check out the article [Managing an App Service Domain ] which provides a detailed steps on how to perform the domain transfer.
Kindly let us know, I’ll follow-up with you further.