Sadly a lot of the allegedly faster (and more efficient) online and phone business you need to do now involves going in circles.
Whether that's a little blue circle happily spinning on a blank web page, or being passed from department to department on the phone . . . once you actually get to speak to a human.
As for DVLA, why not write to them at Swansea? Explain the email address they have is no longer relevant and (if you have moved house) your new address etc.
I'd also mention the problems you've had with their 'system' and suggest an overhaul and some better 'customer service' training
