So I would like to ask if I can keep the old url and add a 302 redirect to a new url.
No, the URL of the project is the permalink.
You can work with 2 versions, a prod one and dev one for example, and switch the name of the current dev one to turn it into a prod, for example.
You can also use a custom domain name which is probably the easiest for your site visitors, but don’t forget to change the name of the custom domain if you switch or remix to another instance/project.