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Try using the route planner here: http://cycle.travel/
It's specifically set up to avoid main roads and to route you via cycle paths or cycle friendly roads.
It can be a little over-cautious at times, but I've used it with great success a number of times.There's a reason why the dual carriageways are so straight! The wiggly roads are the quietest and most picturesque. If you have a decent second battery you can use Navfree or similar to navigate, it uses open street maps uploaded to the phone so doesn't need a data connection. It works very well and you can upload a gps route to it - but it will eat your battery!
Otherwise, the cheapest Garmin etrex is only about £130.
Dear mini-tourers, I'm after some route suggestions to get from Brocklehurst in the New Forest to Cobham in Surrey. Keeping off dual carriageways appears to involve a helluva lot of twists and turns and I'm concerned I'll spend half the day stopping to look at Google Maps on my phone. Preferably avoiding hills but I'm open to ideas.