I would avoid the major roads. Get a map of the Sustrans route (from them – it covers the whole route). You’ll be able to work out road alternatives to the tracks.
The route is designed to be quiet, attractive and fun to ride. The major roads would be hours of hell with risk of death. I’d stick to the Sustrans as far as possible. Lots of people were riding it on roadbikes (hell, I even met one on the old coach road!), but I can’t vouch as to their quality. It will add to the distance a bit.
That said, you might find tourer and tracks ok. It really is a VERY EASY RIDE, apart from some long slog hills in the middle. If you’re used to ‘proper’ biking, it’s a doddle.