Please make sure that the Steps custom field is enabled for the new project. You can do this in the administration area under Administration > Customizations > Case Fields > edit. In the project list above the Save Field button, you can see the projects this custom field applies to. It’s likely that your project is not included in this list. The easiest way to enable the custom field for your new project is to choose the option “These options apply to all projects”.
Regarding the IDs: if you move the test cases instead of copying (Move button on the Copy/Move Test Cases dialog), the IDs are not changed and you don’t need to update your test cases.
I would also recommend making a fresh backup of your database before making further changes, just in case.
I hope this helps and please let me know in case you have any further questions.