That sounds like it’s expected. If you haven’t edited your DNS zone file (which you should not do until you’re ready to cut over) the DNS records will show what’s currently live, which should be your Exchange server.
Running it as a secondary mail server will be more complex than replacing it because they’re two completely separate architectures.
What did you name your mailcow hostname? Try adding a subdomain (or a spare domain, if you have one) and point the MX records of that subdomain to your mailcow instance, you should be able to receive emails.
Once you test that, you’ll need to add the domain(s) to mailcow, add all your users, aliases (aka Distribution Lists) and then use the sync jobs to move the email across. Only then, you should move your MX on the production domain.