Use:
Admin Level -> Admin Backup/Transfer
Related guide:
http://help.directadmin.com/item.php?id=57
If you want to swap the IPs *all* users/zones on the old box (eg: if the box will no longer be used once the transfer is done).. then you can use the ipswap.sh script on the old box, to update everything.
cd /usr/local/directadmin/scripts
./ipswap.sh 1.2.3.4 4.3.2.1
which would swap everything from 1.2.3.4 to 4.3.2.1
Or... just use a perl regex to just change the dns:
cd /var/named
perl -pi -e 's/1.2.3.4/4.3.2.1/' *.db.