We are going through a migration from the self-hosted to the cloud service, and I want to know if the DSN changes during this process.

Yes, you need to swap the already existing DSNs in your Sentry.init() to the new ones you receive after migration, because the older ones can't point to SaaS Sentry. You can also find the DSN in your project settings by navigating to [Project] > Settings > Client Keys (DSN).