Alright, well it's costing me $4/month for ionos to register the domain.
That's top-end but not unheard of (for a .com). Given who ionos are (ie. the provider formerly known as 1and1) then that's outrageous and if I were you I would move. Just make sure that you haven't accidentally signed up for some additional thing which is where the money is really going. For comparison, $WORK provides .com registration and management for £25 p.a. (with a discount on that for multiple years and other bulk purchases) which equates to less than $3 a month at current rates and they are at the quality end of the market (eg. you can phone them up and talk to a technically competent person, etc.)
What are DNS zone settings?
They are referring to the zone data which is essentially your DNS records. This is reasonable in that if your DNS is up and running on some other provider there is no need for your previous provider to retain that. Do check that your new provider is actively serving the DNS and has all the data which from a cursory check appears currently not to be the case:
$ host -t soa merrillpjensen.com.
Host merrillpjensen.com not found: 2(SERVFAIL)