I wouldn't choose to ride the A59 either (it's bad enough just crossing it!) but there are plenty of quieter roads that you could use. I'll work out an alternative route for you. (Time passes... slowly).
Okay, how about:
- (B6243)
- Grimsargh
- Longridge
- Knowle Green
- Hurst Green
- Mitton Green
- (Unclassified roads)
- Bashall Town
- Waddington
- West Bradford
- Grindleton
- Sawley
- Holden
- Forest Becks
- Mere Syke
- Wigglesworth
- Rathmel
- Giggleswick
- Stackhouse
- Little Stainforth
- (B6479)
- Helwith Bridge
- Horton-in-Ribblesdale
The only A-road you'd encounter would be a few yards of the A65 as you cross it just before Giggleswick/Settle.
Here's a rough map: