you are confusing DNS hosting and Web/email hosting
You can renew the DNS hosting with the current registrar and keep your DNS zone over there and move your web hosting to the new provider.
1) what you will need to do manually if you change only your web hosting
change your DNS records to match the new IP address of your new host (www, MX etc...)
2) if you are also changing email hosting then yea for sure you will need to re-create and/or migrate somehow all your email accounts
3) so you have no problem here