Hi,
If this is urgent (I assume it is since your site is down) I recommend you create a Severity A (<1 hour response) Technical support request so that backend engineer can fix it. To create Technical support request in the Azure portal you need to have at least Standard level support. Link below will allow you to upgrade if necessary:
https://portal.azure.com/#view/Microsoft_Azure_Support/HelpAndSupportBlade/~/supportPlans
Create an Azure support request
Creating support request can be confusing so please add a comment if you get stuck in endless loop and I will assist.
If you want to try to fix it in the meantime, you can navigate to your App Service Domain in the portal, click Manage DNS records, click Delegate to Azure DNS, select your existing Azure DNS, etc. This might eventually make the necessary change to GoDaddy, however, it is entirely possible that the process will fail. I believe in the cases where it used to work the change might take time to take effect.
Your ability to manage/update your domain is severely limited, which is why I recommended you create Severity A Technical support request above, since Azure support engineer can actually make changes to your GoDaddy record.
Since there are so many problems and limitations with App Service Domains I no longer recommend people use them if at all possible. It doesn't help your immediate issue, but please consider transferring your domain to another registrar as soon as possible. You will save yourself a lot of future headaches.
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.
You can transfer to a non-GoDaddy registrar, and then if you want wait 60 days and transfer to your existing GoDaddy account.
Please click Accept Answer and upvote if the above was helpful.
Thanks.
-TP