It looks like your new MySQL server is attempting to start at the beginning again for it's auto increment which will cause conflict with existing IDs. You'll need to find the last ID used in each table and that has an auto increment ID and then use:
ALTER TABLE table_name AUTO_INCREMENT = start_value; where
start_value is replaced by the next usable integer and the
table_name is replaced by the table your updating.
I'm unsure why MySQL did this as usually when restoring from backups or updating a database MySQL already keeps this data but it appears it got lost somehow.