You have to change the public record and have the new ISP create the zone
file.
But, I would first add the new ISP DNS to the public record, then have them
draw a secondary zone. This will prevent your domain from being down during
transition.
If you want to drop the previous ISP DNS change the public record and
contact the new ISP to convert the zone to a primary and fix the records to
reflect the correct Primary name server on the SOA record.