Frankly, the question makes no sense.
Did you move your website from one hosting to provider to another? Then you need to log into your domain registrar and point the name server entries to the name servers provided by your new hosting provider.
Did you change the server name on the same server and keep the domain name the same? Then you need to go your cPanel (or whatever else is being used) and update your DNS entries and then wait for your hosting provider to pick up the changed information and pull it into their name servers.
what code (or header) should I use in my old server to tell the visitor's browser or ISP that it should check for my new DNS
This specifically is not allowed as a security measure, unless you actually are asking about a 301 redirect from the old server to the new server which is a HTTP server function, not DNS.