You won't get (nor do you need SSN) to begin work. You don't need new DL to work either. TN and passport will do.
The border used to give a TN a couple of weeks before start, but now often insist that you be starting immediately. If your TN letter has a start date of June 4, don't expect to get TN before June 1.
As to SSN, you can't aply for it until you actaully get TN, and it will not be granted to you until they verify your TN in a databse -- 3-6 weeks.
But you can start working immediately. You likley have 30 days after move to change your plates and DL.