Hi Carl,
I recommend you first transfer your App Service Domain to a different (not GoDaddy or Cloudflare) registrar. Once transfer is complete you can update name servers to point to Cloudflare, and after 60 days you can transfer the domain to Cloudflare.
There used to be workarounds to provide access to DCC to make these types of changes, but they no longer work.
To transfer to different registrar you will need authorization code. You may open Azure Cloud Shell and execute command similar to below from App Service team blog (replace subscription id, resource group, domain name):
Invoke-AzRestMethod -Path "/subscriptions/<SUBSCRIPTION-ID>/resourceGroups/<RESOURCE-GROUP-NAME>/providers/Microsoft.DomainRegistration/domains/<DOMAIN-NAME>/transferout?api-version=2021-02-01" -Method PUT
In the output from the command, you need authCode, which is what you will provide the new domain registrar when requesting transfer in.
Please click Accept Answer and upvote if the above was helpful.
Thanks.
-TP