Are you able to save other WC settings or is this problem only with the “New Order” email setting? Also can you save other WP settings (i.e Outside of WC)? Finally, have you checked to make sure you have the latest version of your theme? Sometimes themes need to be updated to synchronise with the latest WP and WC.
Maybe worth trying the database repair option at
https://www.<yoursite>.com/wp-admin/maint/repair.php
follow the on screen instructions.
If you have installed new plugins this also may be causing some issues with saving the data. You may also want to consider deactivating and deleting the WooCommerce plugin and reinstalling from a fresh download. Your settings and products should be in the database and, if so, will not be affected.
Can you post a screenshot of your WC settings as it relates to the “new order” – there may be some clues there that could help.
