The best way to tax your car will be with a trip to your local car tax issuing Post Office. You can take your old-style V5 with you (have the change of address section filled in). You should be able to tax it there and then as long as it is insured (it doesn't need an MoT as it built before 1960), and retain the V5 to send to the DVLA for the change of address. If for any reason they won't accept the old V5, you can still tax the car using a form V62 - which means you're applying for a new V5C in your name at your current address.