Hi Michael,
Transferring domain in is currently not supported. See excerpt from Azure App Service team blog article below:
Transfering of domain into Azure is currently not supported. Any attempts made to transfer the domain into Azure risks getting the domain into a broken state. We will not be providing support to fix domains caught in this broken state due to unsupported transfer in scenario.
If you want, you could delegate DNS zone to Azure DNS so that you could manage DNS records within Azure portal, but from your description I understand you are wanting more than that. Plus, if you are using Cloudflare services for the domain you likely need to keep DNS with them.
Please click Accept Answer and upvote if the above was helpful.
Thanks.
-TP