Wanted to give everyone a heads up, supervisor is being updated now. 2022.05.0 is being pushed out to stable which includes the option to disable the fallback DNS. Additionally both SSH addons (the official one and the one in community) have been updated with the latest CLI version that includes support for disabling the fallback. Which means you can ignore all the curl stuff above and just type this:
ha dns options --fallback=false
Although I would recommend running ha resolution info first to make sure that supervisor did not identify any issues with your configured DNS servers. These checks do run on a schedule so if you want to be sure the check ran you can either wait a couple hours or force it to run immediately by doing ha resolution healthcheck.
One minor catch, the DNS plugin also needs an update to acknowledge and support this new option. We’re going to wait about 8 hours from now to merge https://github.com/home-assistant/version/pull/223 so that supervisor has updated everywhere first. Once this second PR goes in and your DNS plugin updates then you’re free to disable the fallback DNS if you want.