You may use the Admin Console to check for your deleted Shared Drive through this link. It will be on the very right side so you'll need to scroll to it.
Note: From what I have experienced, deleted shared drives will be on this page for up to 25 days.
You should be able to get SharedDrives ID through the Method: drives.list API.
But for some instances, the API only gets shared drives that are active and not deleted. With this, you may submit a Feature Request on this link to inform Google about this
issue.