Trevor,
The A169 has some steep drops and climbs especially at the Hole of Horcum area. I was at the CC site at West Ayton Nr Scarborough and had a ride out as far as Goathland along the A169 and it would be the very last resort for me personally to tow along that road. Most of it is open heath up to this point. Can't help on the stretch from Goathland to Whitby has its 25yrs since I did that stretch.
The A64 is a good road, with some duel carrage ways, all the way to Scarborough and If you decide on that route, stay on this and ignore the A170 and head for scarborough on the A64. You will then almost join the A171 at Scarborough for Whitby. Having traveled along the A171 a few times on my visit to Whitby, this is the better route for towing, but this also needs extra care in places. Don,t be duped by the shorter route accross the moors although I did see the very occasional brave soul who was towing. If you do decide on that route, make sure its dry and clear as there are some steep and sharp corners.
Have a nice time and take care.
Brian